Mathematics NCERT Grade 10, Chapter 6 - Triangles: Everyone is familiar with triangles. Factors like congruent and similar are known to students. But looking more into the same a short description of similar figures is given.
• It is concluded that all congruent figures are similar but similar figures need not be congruent.
Certain factors are stated that helps in recognizing whether the figures are similar or not
• Activity given lays emphasis on that two polygons of the same number of sides are similar if:
a. all the corresponding angles are equal.
b. all the corresponding sides are in the same ratio.

The next section is about the topic- Similarity of Triangles.
• Two triangles are similar if (i) their corresponding angles are equal. (ii) their corresponding sides are in the same ratio.
Several theorems and activities are also given in this section. basic proportionality theorem must be studied which states that if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio. Its converse also needs to be studied. This is followed by important examples and exercise questions based on the same concept.
The next sections deal with different criteria that are used to prove the similarity of triangles. Criteria listed in this section:

• AAA(Angle - Angle - Angle) Similarity criterion
• AA(Angle - Angle) Similarity criterion
• SSS(Side - Side - Side) similarity criterion
• SAS(Side - Angle - Side) Similarity criterion
In the next exercise, students have to state whether the given triangles are similar or not and some question requires proves of certain criteria. Exercise 6.4 is based on the topic- Areas of similar triangles according to which the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.
Next section deals with Pythagoras theorem. Also, three theorems are provided which will help the students to prove the Pythagoras theorem. The proof of the Pythagoras theorem is important from the examination point of view. Next exercise is based on the same.
In the end, there is an optional exercise followed by summarization of key points.

#### Question 1:

Fill in the blanks using correct word given in the brackets:−

(i) All circles are __________. (congruent, similar)

(ii) All squares are __________. (similar, congruent)

(iii) All __________ triangles are similar. (isosceles, equilateral)

(iv) Two polygons of the same number of sides are similar, if (a) their corresponding angles are __________ and (b) their corresponding sides are __________. (equal, proportional)

#### Answer:

(i) Similar

(ii) Similar

(iii) Equilateral

(iv) (a) Equal

(b) Proportional

##### Video Solution for triangles (Page: 122 , Q.No.: 1)

NCERT Solution for Class 10 math - triangles 122 , Question 1

#### Question 2:

Give two different examples of pair of

(i) Similar figures

(ii)Non-similar figures

#### Answer:

(i) Two equilateral triangles with sides 1 cm and 2 cm

Two squares with sides 1 cm and 2 cm

(ii) Trapezium and square

Triangle and parallelogram

##### Video Solution for triangles (Page: 122 , Q.No.: 2)

NCERT Solution for Class 10 math - triangles 122 , Question 2

#### Question 1:

In figure.6.17. (i) and (ii), DE || BC. Find EC in (i) and AD in (ii).

(i)

(ii)

#### Answer:

(i)

Let EC = x cm

It is given that DE || BC.

By using basic proportionality theorem, we obtain

(ii)

Let AD = x cm

It is given that DE || BC.

By using basic proportionality theorem, we obtain

##### Video Solution for triangles (Page: 128 , Q.No.: 1)

NCERT Solution for Class 10 math - triangles 128 , Question 1

#### Question 2:

E and F are points on the sides PQ and PR respectively of a ΔPQR. For each of the following cases, state whether EF || QR.

(i) PE = 3.9 cm, EQ = 3 cm, PF = 3.6 cm and FR = 2.4 cm

(ii) PE = 4 cm, QE = 4.5 cm, PF = 8 cm and RF = 9 cm

(iii)PQ = 1.28 cm, PR = 2.56 cm, PE = 0.18 cm and PF = 0.63 cm

#### Answer:

(i)

Given that, PE = 3.9 cm, EQ = 3 cm, PF = 3.6 cm, FR = 2.4 cm

(ii)

PE = 4 cm, QE = 4.5 cm, PF = 8 cm, RF = 9 cm

(iii)

PQ = 1.28 cm, PR = 2.56 cm, PE = 0.18 cm, PF = 0.36 cm

##### Video Solution for triangles (Page: 128 , Q.No.: 2)

NCERT Solution for Class 10 math - triangles 128 , Question 2

#### Question 3:

In the following figure, if LM || CB and LN || CD, prove that

#### Answer:

In the given figure, LM || CB

By using basic proportionality theorem, we obtain

##### Video Solution for triangles (Page: 128 , Q.No.: 3)

NCERT Solution for Class 10 math - triangles 128 , Question 3

#### Question 4:

In the following figure, DE || AC and DF || AE. Prove that

#### Answer:

In ΔABC, DE || AC

##### Video Solution for triangles (Page: 128 , Q.No.: 4)

NCERT Solution for Class 10 math - triangles 128 , Question 4

#### Question 5:

In the following figure, DE || OQ and DF || OR, show that EF || QR.

#### Answer:

In Δ POQ, DE || OQ

##### Video Solution for triangles (Page: 129 , Q.No.: 5)

NCERT Solution for Class 10 math - triangles 129 , Question 5

#### Question 6:

In the following figure, A, B and C are points on OP, OQ and OR respectively such that AB || PQ and AC || PR. Show that BC || QR.

#### Answer:

In Δ POQ, AB || PQ

##### Video Solution for triangles (Page: 129 , Q.No.: 6)

NCERT Solution for Class 10 math - triangles 129 , Question 6

#### Question 7:

Using Basic proportionality theorem, prove that a line drawn through the mid-points of one side of a triangle parallel to another side bisects the third side. (Recall that you have proved it in Class IX).

#### Answer:

Consider the given figure in which l is a line  drawn through the mid-point P of line segment AB meeting AC at Q, such that .

Or, Q is the mid-point of AC.

##### Video Solution for triangles (Page: 129 , Q.No.: 7)

NCERT Solution for Class 10 math - triangles 129 , Question 7

#### Question 8:

Using Converse of basic proportionality theorem, prove that the line joining the mid-points of any two sides of a triangle is parallel to the third side. (Recall that you have done it in Class IX).

#### Answer:

Consider the given figure in which PQ is a line segment joining the mid-points P and Q of line AB and AC respectively.

i.e., AP = PB and AQ = QC

It can be observed that

Hence, by using basic proportionality theorem, we obtain

##### Video Solution for triangles (Page: 129 , Q.No.: 8)

NCERT Solution for Class 10 math - triangles 129 , Question 8

#### Question 9:

ABCD is a trapezium in which AB || DC and its diagonals intersect each other at the point O. Show that

#### Answer:

Draw a line EF through point O, such that

In ΔADC,

By using basic proportionality theorem, we obtain

In ΔABD,

So, by using basic proportionality theorem, we obtain

From equations (1) and (2), we obtain

##### Video Solution for triangles (Page: 129 , Q.No.: 9)

NCERT Solution for Class 10 math - triangles 129 , Question 9

#### Question 10:

The diagonals of a quadrilateral ABCD intersect each other at the point O such that Show that ABCD is a trapezium.

#### Answer:

Let us consider the following figure for the given question.

Draw a line OE || AB

In ΔABD, OE || AB

By using basic proportionality theorem, we obtain

However, it is given that

⇒ EO || DC [By the converse of basic proportionality theorem]

⇒ AB || OE || DC

⇒ AB || CD

∴ ABCD is a trapezium.

##### Video Solution for triangles (Page: 129 , Q.No.: 10)

NCERT Solution for Class 10 math - triangles 129 , Question 10

#### Question 1:

State which pairs of triangles in the following figure are similar? Write the similarity criterion used by you for answering the question and also write the pairs of similar triangles in the symbolic form:

(i)

(ii)

(iii)

(iv)

(v)

(vi)

#### Answer:

(i) ∠A = ∠P = 60°

∠B = ∠Q = 80°

∠C = ∠R = 40°

Therefore, ΔABC ∼ ΔPQR [By AAA similarity criterion]

(ii)

(iii)The given triangles are not similar as the corresponding sides are not proportional.

(iv) In ∆ MNL and ∆ QPR, we observe that,

(v)The given triangles are not similar as the corresponding sides are not proportional.

(vi) In ΔDEF,

∠D +∠E +∠F = 180º

(Sum of the measures of the angles of a triangle is 180º.)

70º + 80º +∠F = 180º

∠F = 30º

Similarly, in ΔPQR,

∠P +∠Q +∠R = 180º

(Sum of the measures of the angles of a triangle is 180º.)

∠P + 80º +30º = 180º

∠P = 70º

In ΔDEF and ΔPQR,

∠D = ∠P (Each 70°)

∠E = ∠Q (Each 80°)

∠F = ∠R (Each 30°)

∴ ΔDEF ∼ ΔPQR [By AAA similarity criterion]

##### Video Solution for triangles (Page: 138 , Q.No.: 1)

NCERT Solution for Class 10 math - triangles 138 , Question 1

#### Question 3:

Diagonals AC and BD of a trapezium ABCD with AB || DC intersect each other at the point O. Using a similarity criterion for two triangles, show that

#### Answer:

In ΔDOC and ΔBOA,

∠CDO = ∠ABO [Alternate interior angles as AB || CD]

∠DCO = ∠BAO [Alternate interior angles as AB || CD]

∠DOC = ∠BOA [Vertically opposite angles]

∴ ΔDOC ∼ ΔBOA [AAA similarity criterion]

##### Video Solution for triangles (Page: 139 , Q.No.: 3)

NCERT Solution for Class 10 math - triangles 139 , Question 3

#### Question 4:

In the following figure, Show that

#### Answer:

In ΔPQR, ∠PQR = ∠PRQ

∴ PQ = PR (i)

Given,

##### Video Solution for triangles (Page: 140 , Q.No.: 4)

NCERT Solution for Class 10 math - triangles 140 , Question 4

#### Question 5:

S and T are point on sides PR and QR of ΔPQR such that ∠P = ∠RTS. Show that ΔRPQ ∼ ΔRTS.

#### Answer:

In ΔRPQ and ΔRST,

∠RTS = ∠QPS (Given)

∠R = ∠R (Common angle)

∴ ΔRPQ ∼ ΔRTS (By AA similarity criterion)

##### Video Solution for triangles (Page: 140 , Q.No.: 5)

NCERT Solution for Class 10 math - triangles 140 , Question 5

#### Question 6:

In the following figure, if ΔABE ≅ ΔACD, show that ΔADE ∼ ΔABC.

#### Answer:

It is given that ΔABE ≅ ΔACD.

∴ AB = AC [By CPCT] (1)

And, AD = AE [By CPCT] (2)

In ΔADE and ΔABC,

[Dividing equation (2) by (1)]

∠A = ∠A [Common angle]

∴ ΔADE ∼ ΔABC [By SAS similarity criterion]

##### Video Solution for triangles (Page: 140 , Q.No.: 6)

NCERT Solution for Class 10 math - triangles 140 , Question 6

#### Question 7:

In the following figure, altitudes AD and CE of ΔABC intersect each other at the point P. Show that:

(i) ΔAEP ∼ ΔCDP

(ii) ΔABD ∼ ΔCBE

(iii) ΔAEP ∼ ΔADB

(v) ΔPDC ∼ ΔBEC

#### Answer:

(i)

In ΔAEP and ΔCDP,

∠AEP = ∠CDP (Each 90°)

∠APE = ∠CPD (Vertically opposite angles)

Hence, by using AA similarity criterion,

ΔAEP ∼ ΔCDP

(ii)

In ΔABD and ΔCBE,

∠ADB = ∠CEB (Each 90°)

∠ABD = ∠CBE (Common)

Hence, by using AA similarity criterion,

ΔABD ∼ ΔCBE

(iii)

In ΔAEP and ΔADB,

∠AEP = ∠ADB (Each 90°)

∠PAE = ∠DAB (Common)

Hence, by using AA similarity criterion,

ΔAEP ∼ ΔADB

(iv)

In ΔPDC and ΔBEC,

∠PDC = ∠BEC (Each 90°)

∠PCD = ∠BCE (Common angle)

Hence, by using AA similarity criterion,

ΔPDC ∼ ΔBEC

##### Video Solution for triangles (Page: 140 , Q.No.: 7)

NCERT Solution for Class 10 math - triangles 140 , Question 7

#### Question 8:

E is a point on the side AD produced of a parallelogram ABCD and BE intersects CD at F. Show that ΔABE ∼ ΔCFB

#### Answer:

In ΔABE and ΔCFB,

∠A = ∠C (Opposite angles of a parallelogram)

∠AEB = ∠CBF (Alternate interior angles as AE || BC)

∴ ΔABE ∼ ΔCFB (By AA similarity criterion)

##### Video Solution for triangles (Page: 140 , Q.No.: 8)

NCERT Solution for Class 10 math - triangles 140 , Question 8

#### Question 9:

In the following figure, ABC and AMP are two right triangles, right angled at B and M respectively, prove that:

(i) ΔABC ∼ ΔAMP

(ii)

#### Answer:

In ΔABC and ΔAMP,

∠ABC = ∠AMP (Each 90°)

∠A = ∠A (Common)

∴ ΔABC ∼ ΔAMP (By AA similarity criterion)

##### Video Solution for triangles (Page: 140 , Q.No.: 9)

NCERT Solution for Class 10 math - triangles 140 , Question 9

#### Question 10:

CD and GH are respectively the bisectors of ∠ACB and ∠EGF such that D and H lie on sides AB and FE of ΔABC and ΔEFG respectively. If ΔABC ∼ ΔFEG, Show that:

(i)

(ii) ΔDCB ∼ ΔHGE

(iii) ΔDCA ∼ ΔHGF

#### Answer:

It is given that ΔABC ∼ ΔFEG.

∴ ∠A = ∠F, ∠B = ∠E, and ∠ACB = ∠FGE

∠ACB = ∠FGE

∴ ∠ACD = ∠FGH (Angle bisector)

And, ∠DCB = ∠HGE (Angle bisector)

In ΔACD and ΔFGH,

∠A = ∠F (Proved above)

∠ACD = ∠FGH (Proved above)

∴ ΔACD ∼ ΔFGH (By AA similarity criterion)

In ΔDCB and ΔHGE,

∠DCB = ∠HGE (Proved above)

∠B = ∠E (Proved above)

∴ ΔDCB ∼ ΔHGE (By AA similarity criterion)

In ΔDCA and ΔHGF,

∠ACD = ∠FGH (Proved above)

∠A = ∠F (Proved above)

∴ ΔDCA ∼ ΔHGF (By AA similarity criterion)

##### Video Solution for triangles (Page: 140 , Q.No.: 10)

NCERT Solution for Class 10 math - triangles 140 , Question 10

#### Question 11:

In the following figure, E is a point on side CB produced of an isosceles triangle ABC with AB = AC. If AD ⊥ BC and EF ⊥ AC, prove that ΔABD ∼ ΔECF

#### Answer:

It is given that ABC is an isosceles triangle.

∴ AB = AC

⇒ ∠ABD = ∠ECF

In ΔABD and ΔECF,

∠ADB = ∠EFC (Each 90°)

∠ABD = ∠ECF (Proved above)

∴ ΔABD ∼ ΔECF (By using AA similarity criterion)

##### Video Solution for triangles (Page: 141 , Q.No.: 11)

NCERT Solution for Class 10 math - triangles 141 , Question 11

#### Question 12:

Sides AB and BC and median AD of a triangle ABC are respectively proportional to sides PQ and QR and median PM of ΔPQR (see the given figure). Show that ΔABC ∼ ΔPQR.

#### Answer:

Median divides the opposite side.

Given that,

In ΔABD and ΔPQM,

(Proved above)

∴ ΔABD ∼ ΔPQM (By SSS similarity criterion)

⇒ ∠ABD = ∠PQM (Corresponding angles of similar triangles)

In ΔABC and ΔPQR,

∠ABD = ∠PQM (Proved above)

∴ ΔABC ∼ ΔPQR (By SAS similarity criterion)

##### Video Solution for triangles (Page: 141 , Q.No.: 12)

NCERT Solution for Class 10 math - triangles 141 , Question 12

#### Question 13:

D is a point on the side BC of a triangle ABC such that ∠ADC = ∠BAC. Show that

#### Answer:

In ΔADC and ΔBAC,

∠ADC = ∠BAC (Given)

∠ACD = ∠BCA (Common angle)

∴ ΔADC ∼ ΔBAC (By AA similarity criterion)

We know that corresponding sides of similar triangles are in proportion.

##### Video Solution for triangles (Page: 141 , Q.No.: 13)

NCERT Solution for Class 10 math - triangles 141 , Question 13

#### Question 14:

Sides AB and AC and median AD of a triangle ABC are respectively proportional to sides PQ and PR and median PM of another triangle PQR. Show that

#### Answer:

Given that,

Let us extend AD and PM up to point E and L respectively, such that AD = DE and PM = ML. Then, join B to E, C to E, Q to L, and R to L.

We know that medians divide opposite sides.

Therefore, BD = DC and QM = MR

Also, AD = DE (By construction)

And, PM = ML (By construction)

In quadrilateral ABEC, diagonals AE and BC bisect each other at point D.

Therefore, quadrilateral ABEC is a parallelogram.

∴ AC = BE and AB = EC (Opposite sides of a parallelogram are equal)

Similarly, we can prove that quadrilateral PQLR is a parallelogram and PR = QL, PQ = LR

It was given that

∴ ΔABE ∼ ΔPQL (By SSS similarity criterion)

We know that corresponding angles of similar triangles are equal.

∴ ∠BAE = ∠QPL … (1)

Similarly, it can be proved that ΔAEC ∼ ΔPLR and

∠CAE = ∠RPL … (2)

Adding equation (1) and (2), we obtain

∠BAE + ∠CAE = ∠QPL + ∠RPL

⇒ ∠CAB = ∠RPQ … (3)

In ΔABC and ΔPQR,

(Given)

∠CAB = ∠RPQ [Using equation (3)]

∴ ΔABC ∼ ΔPQR (By SAS similarity criterion)

##### Video Solution for triangles (Page: 141 , Q.No.: 14)

NCERT Solution for Class 10 math - triangles 141 , Question 14

#### Question 15:

A vertical pole of a length 6 m casts a shadow 4m long on the ground and at the same time a tower casts a shadow 28 m long. Find the height of the tower.

#### Answer:

Let AB and CD be a tower and a pole respectively.

Let the shadow of BE and DF be the shadow of AB and CD respectively.

At the same time, the light rays from the sun will fall on the tower and the pole at the same angle.

Therefore, ∠DCF = ∠BAE

And, ∠DFC = ∠BEA

∠CDF = ∠ABE (Tower and pole are vertical to the ground)

∴ ΔABE ∼ ΔCDF (AAA similarity criterion)

Therefore, the height of the tower will be 42 metres.

##### Video Solution for triangles (Page: 141 , Q.No.: 15)

NCERT Solution for Class 10 math - triangles 141 , Question 15

#### Question 16:

If AD and PM are medians of triangles ABC and PQR, respectively where

#### Answer:

It is given that ΔABC ∼ ΔPQR

We know that the corresponding sides of similar triangles are in proportion.

… (1)

Also, ∠A = ∠P, ∠B = ∠Q, ∠C = ∠R … (2)

Since AD and PM are medians, they will divide their opposite sides.

… (3)

From equations (1) and (3), we obtain

… (4)

In ΔABD and ΔPQM,

∠B = ∠Q [Using equation (2)]

[Using equation (4)]

∴ ΔABD ∼ ΔPQM (By SAS similarity criterion)

##### Video Solution for triangles (Page: 141 , Q.No.: 16)

NCERT Solution for Class 10 math - triangles 141 , Question 16

#### Question 1:

Let and their areas be, respectively, 64 cm2 and 121 cm2. If EF = 15.4 cm, find BC.

#### Answer:

##### Video Solution for triangles (Page: 143 , Q.No.: 1)

NCERT Solution for Class 10 math - triangles 143 , Question 1

#### Question 2:

Diagonals of a trapezium ABCD with AB || DC intersect each other at the point O. If AB = 2CD, find the ratio of the areas of triangles AOB and COD.

#### Answer:

Since AB || CD,

∴ ∠OAB = ∠OCD and ∠OBA = ∠ODC (Alternate interior angles)

In ΔAOB and ΔCOD,

∠AOB = ∠COD (Vertically opposite angles)

∠OAB = ∠OCD (Alternate interior angles)

∠OBA = ∠ODC (Alternate interior angles)

∴ ΔAOB ∼ ΔCOD (By AAA similarity criterion)

##### Video Solution for triangles (Page: 143 , Q.No.: 2)

NCERT Solution for Class 10 math - triangles 143 , Question 2

#### Question 3:

In the following figure, ABC and DBC are two triangles on the same base BC. If AD intersects BC at O, show that

#### Answer:

Let us draw two perpendiculars AP and DM on line BC.

We know that area of a triangle =

.

In ΔAPO and ΔDMO,

∠APO = ∠DMO (Each = 90°)

∠AOP = ∠DOM (Vertically opposite angles)

∴ ΔAPO ∼ ΔDMO (By AA similarity criterion)

##### Video Solution for triangles (Page: 144 , Q.No.: 3)

NCERT Solution for Class 10 math - triangles 144 , Question 3

#### Question 4:

If the areas of two similar triangles are equal, prove that they are congruent.

#### Answer:

Let us assume two similar triangles as ΔABC ∼ ΔPQR.

##### Video Solution for triangles (Page: 144 , Q.No.: 4)

NCERT Solution for Class 10 math - triangles 144 , Question 4

#### Question 5:

D, E and F are respectively the mid-points of sides AB, BC and CA of ΔABC. Find the ratio of the area of ΔDEF and ΔABC.

#### Answer:

D and E are the mid-points of ΔABC.

##### Video Solution for triangles (Page: 144 , Q.No.: 5)

NCERT Solution for Class 10 math - triangles 144 , Question 5

#### Question 6:

Prove that the ratio of the areas of two similar triangles is equal to the square

of the ratio of their corresponding medians.

#### Answer:

Let us assume two similar triangles as ΔABC ∼ ΔPQR. Let AD and PS be the medians of these triangles.

ΔABC ∼ ΔPQR

…(1)

∠A = ∠P, ∠B = ∠Q, ∠C = ∠R … (2)

Since AD and PS are medians,

∴ BD = DC =

And, QS = SR =

Equation (1) becomes

… (3)

In ΔABD and ΔPQS,

∠B = ∠Q [Using equation (2)]

And, [Using equation (3)]

∴ ΔABD ∼ ΔPQS (SAS similarity criterion)

Therefore, it can be said that

… (4)

From equations (1) and (4), we may find that

And hence,

##### Video Solution for triangles (Page: 144 , Q.No.: 6)

NCERT Solution for Class 10 math - triangles 144 , Question 6

#### Question 7:

Prove that the area of an equilateral triangle described on one side of a square is equal to half the area of the equilateral triangle described on one of its diagonals.

#### Answer:

Let ABCD be a square of side a.

Therefore, its diagonal

Two desired equilateral triangles are formed as ΔABE and ΔDBF.

Side of an equilateral triangle, ΔABE, described on one of its sides = a

Side of an equilateral triangle, ΔDBF, described on one of its diagonals

We know that equilateral triangles have all its angles as 60º and all its sides of the same length. Therefore, all equilateral triangles are similar to each other. Hence, the ratio between the areas of these triangles will be equal to the square of the ratio between the sides of these triangles.

##### Video Solution for triangles (Page: 144 , Q.No.: 7)

NCERT Solution for Class 10 math - triangles 144 , Question 7

#### Question 8:

ABC and BDE are two equilateral triangles such that D is the mid-point of BC. Ratio of the area of triangles ABC and BDE is

(A) 2 : 1

(B) 1 : 2

(C) 4 : 1

(D) 1 : 4

#### Answer:

We know that equilateral triangles have all its angles as 60º and all its sides of the same length. Therefore, all equilateral triangles are similar to each other. Hence, the ratio between the areas of these triangles will be equal to the square of the ratio between the sides of these triangles.

Let side of ΔABC = x

Therefore, side of

Hence, the correct answer is (C).

##### Video Solution for triangles (Page: 144 , Q.No.: 8)

NCERT Solution for Class 10 math - triangles 144 , Question 8

#### Question 9:

Sides of two similar triangles are in the ratio 4 : 9. Areas of these triangles are in the ratio

(A) 2 : 3

(B) 4 : 9

(C) 81 : 16

(D) 16 : 81

#### Answer:

If two triangles are similar to each other, then the ratio of the areas of these triangles will be equal to the square of the ratio of the corresponding sides of these triangles.

It is given that the sides are in the ratio 4:9.

Therefore, ratio between areas of these triangles =

Hence, the correct answer is (D).

##### Video Solution for triangles (Page: 144 , Q.No.: 9)

NCERT Solution for Class 10 math - triangles 144 , Question 9

#### Question 1:

Sides of triangles are given below. Determine which of them are right triangles? In case of a right triangle, write the length of its hypotenuse.

(i) 7 cm, 24 cm, 25 cm

(ii) 3 cm, 8 cm, 6 cm

(iii) 50 cm, 80 cm, 100 cm

(iv) 13 cm, 12 cm, 5 cm

#### Answer:

(i) It is given that the sides of the triangle are 7 cm, 24 cm, and 25 cm.

Squaring the lengths of these sides, we will obtain 49, 576, and 625.

49 + 576 = 625

Or,

The sides of the given triangle are satisfying Pythagoras theorem.

Therefore, it is a right triangle.

We know that the longest side of a right triangle is the hypotenuse.

Therefore, the length of the hypotenuse of this triangle is 25 cm.

(ii) It is given that the sides of the triangle are 3 cm, 8 cm, and 6 cm.

Squaring the lengths of these sides, we will obtain 9, 64, and 36.

However, 9 + 36 ≠ 64

Or, 32 + 62 ≠ 82

Clearly, the sum of the squares of the lengths of two sides is not equal to the square of the length of the third side.

Therefore, the given triangle is not satisfying Pythagoras theorem.

Hence, it is not a right triangle.

(iii)Given that sides are 50 cm, 80 cm, and 100 cm.

Squaring the lengths of these sides, we will obtain 2500, 6400, and 10000.

However, 2500 + 6400 ≠ 10000

Or, 502 + 802 ≠ 1002

Clearly, the sum of the squares of the lengths of two sides is not equal to the square of the length of the third side.

Therefore, the given triangle is not satisfying Pythagoras theorem.

Hence, it is not a right triangle.

(iv)Given that sides are 13 cm, 12 cm, and 5 cm.

Squaring the lengths of these sides, we will obtain 169, 144, and 25.

Clearly, 144 +25 = 169

Or,

The sides of the given triangle are satisfying Pythagoras theorem.

Therefore, it is a right triangle.

We know that the longest side of a right triangle is the hypotenuse.

Therefore, the length of the hypotenuse of this triangle is 13 cm.

##### Video Solution for triangles (Page: 150 , Q.No.: 1)

NCERT Solution for Class 10 math - triangles 150 , Question 1

#### Question 2:

PQR is a triangle right angled at P and M is a point on QR such that PM ⊥ QR. Show that PM2 = QM × MR.

#### Answer:

##### Video Solution for triangles (Page: 150 , Q.No.: 2)

NCERT Solution for Class 10 math - triangles 150 , Question 2

#### Question 3:

In the following figure, ABD is a triangle right angled at A and AC ⊥ BD. Show that

(i) AB2 = BC × BD

(ii) AC2 = BC × DC

(iii) AD2 = BD × CD

#### Answer:

(i) In ,

(AA similarity criterion)

(ii)

(iii)

∠DCA = ∠ DAB (Each 90º)

∠CDA = ∠ ADB (Common angle)

##### Video Solution for triangles (Page: 150 , Q.No.: 3)

NCERT Solution for Class 10 math - triangles 150 , Question 3

#### Question 4:

ABC is an isosceles triangle right angled at C. prove that AB2 = 2 AC2.

#### Answer:

Given that ΔABC is an isosceles triangle.

∴ AC = CB

Applying Pythagoras theorem in ΔABC (i.e., right-angled at point C), we obtain

##### Video Solution for triangles (Page: 150 , Q.No.: 4)

NCERT Solution for Class 10 math - triangles 150 , Question 4

#### Question 5:

ABC is an isosceles triangle with AC = BC. If AB2 = 2 AC2, prove that ABC is a right triangle.

#### Answer:

Given that,

##### Video Solution for triangles (Page: 150 , Q.No.: 5)

NCERT Solution for Class 10 math - triangles 150 , Question 5

#### Question 6:

ABC is an equilateral triangle of side 2a. Find each of its altitudes.

#### Answer:

Let AD be the altitude in the given equilateral triangle, ΔABC.

We know that altitude bisects the opposite side.

∴ BD = DC = a

In an equilateral triangle, all the altitudes are equal in length.

Therefore, the length of each altitude will be.

##### Video Solution for triangles (Page: 150 , Q.No.: 6)

NCERT Solution for Class 10 math - triangles 150 , Question 6

#### Question 7:

Prove that the sum of the squares of the sides of rhombus is equal to the sum of the squares of its diagonals.

#### Answer:

In ΔAOB, ΔBOC, ΔCOD, ΔAOD,

Applying Pythagoras theorem, we obtain

##### Video Solution for triangles (Page: 150 , Q.No.: 7)

NCERT Solution for Class 10 math - triangles 150 , Question 7

#### Question 8:

In the following figure, O is a point in the interior of a triangle ABC, OD ⊥ BC, OE ⊥ AC and OF ⊥ AB. Show that

(i) OA2 + OB2 + OC2 − OD2 − OE2 − OF2 = AF2 + BD2 + CE2

(ii) AF2 + BD2 + CE2 = AE2 + CD2 + BF2

#### Answer:

Join OA, OB, and OC.

(i) Applying Pythagoras theorem in ΔAOF, we obtain

Similarly, in ΔBOD,

Similarly, in ΔCOE,

(ii) From the above result,

##### Video Solution for triangles (Page: 151 , Q.No.: 8)

NCERT Solution for Class 10 math - triangles 151 , Question 8

#### Question 9:

A ladder 10 m long reaches a window 8 m above the ground. Find the distance of the foot of the ladder from base of the wall.

#### Answer:

Let OA be the wall and AB be the ladder.

Therefore, by Pythagoras theorem,

Therefore, the distance of the foot of the ladder from the base of the wall is 6 m.

##### Video Solution for triangles (Page: 151 , Q.No.: 9)

NCERT Solution for Class 10 math - triangles 151 , Question 9

#### Question 10:

A guy wire attached to a vertical pole of height 18 m is 24 m long and has a stake attached to the other end. How far from the base of the pole should the stake be driven so that the wire will be taut?

#### Answer:

Let OB be the pole and AB be the wire.

By Pythagoras theorem,

Therefore, the distance from the base is m.

##### Video Solution for triangles (Page: 151 , Q.No.: 10)

NCERT Solution for Class 10 math - triangles 151 , Question 10

#### Question 11:

An aeroplane leaves an airport and flies due north at a speed of 1,000 km per hour. At the same time, another aeroplane leaves the same airport and flies due west at a speed of 1,200 km per hour. How far apart will be the two planes after hours?

#### Answer:

Distance travelled by the plane flying towards north in

Similarly, distance travelled by the plane flying towards west in

Let these distances be represented by OA and OB respectively.

Applying Pythagoras theorem,

Distance between these planes after, AB =

Therefore, the distance between these planes will be km after.

##### Video Solution for triangles (Page: 151 , Q.No.: 11)

NCERT Solution for Class 10 math - triangles 151 , Question 11

#### Question 12:

Two poles of heights 6 m and 11 m stand on a plane ground. If the distance between the feet of the poles is 12 m, find the distance between their tops.

#### Answer:

Let CD and AB be the poles of height 11 m and 6 m.

Therefore, CP = 11 − 6 = 5 m

From the figure, it can be observed that AP = 12m

Applying Pythagoras theorem for ΔAPC, we obtain

Therefore, the distance between their tops is 13 m.

##### Video Solution for triangles (Page: 151 , Q.No.: 12)

NCERT Solution for Class 10 math - triangles 151 , Question 12

#### Question 13:

D and E are points on the sides CA and CB respectively of a triangle ABC right angled at C. Prove that AE2 + BD2 = AB2 + DE2

#### Answer:

Applying Pythagoras theorem in ΔACE, we obtain

##### Video Solution for triangles (Page: 151 , Q.No.: 13)

NCERT Solution for Class 10 math - triangles 151 , Question 13

#### Question 14:

The perpendicular from A on side BC of a ΔABC intersect BC at D such that DB = 3 CD. Prove that 2 AB2 = 2 AC2 + BC2

#### Answer:

Applying Pythagoras theorem for ΔACD, we obtain

Applying Pythagoras theorem in ΔABD, we obtain

##### Video Solution for triangles (Page: 151 , Q.No.: 14)

NCERT Solution for Class 10 math - triangles 151 , Question 14

#### Question 15:

In an equilateral triangle ABC, D is a point on side BC such that BD = BC. Prove that 9 AD2 = 7 AB2.

#### Answer:

Let the side of the equilateral triangle be a, and AE be the altitude of ΔABC.

∴ BE = EC = =

And, AE =

Given that, BD = BC

∴ BD =

DE = BE − BD =

Applying Pythagoras theorem in ΔADE, we obtain

AD2 = AE2 + DE2

⇒ 9 AD2 = 7 AB2

##### Video Solution for triangles (Page: 151 , Q.No.: 15)

NCERT Solution for Class 10 math - triangles 151 , Question 15

#### Question 16:

In an equilateral triangle, prove that three times the square of one side is equal to four times the square of one of its altitudes.

#### Answer:

Let the side of the equilateral triangle be a, and AE be the altitude of ΔABC.

∴ BE = EC = =

Applying Pythagoras theorem in ΔABE, we obtain

AB2 = AE2 + BE2

4AE2 = 3a2

⇒ 4 × (Square of altitude) = 3 × (Square of one side)

##### Video Solution for triangles (Page: 151 , Q.No.: 16)

NCERT Solution for Class 10 math - triangles 151 , Question 16

#### Question 17:

Tick the correct answer and justify: In ΔABC, AB = cm, AC = 12 cm and BC = 6 cm.

The angle B is:

(A) 120° (B) 60°

(C) 90° (D) 45°

#### Answer:

Given that, AB =cm, AC = 12 cm, and BC = 6 cm

It can be observed that

AB2 = 108

AC2 = 144

And, BC2 = 36

AB2 +BC2 = AC2

The given triangle, ΔABC, is satisfying Pythagoras theorem.

Therefore, the triangle is a right triangle, right-angled at B.

∴ ∠B = 90°

Hence, the correct answer is (C).

##### Video Solution for triangles (Page: 151 , Q.No.: 17)

NCERT Solution for Class 10 math - triangles 151 , Question 17

#### Question 1:

In the given figure, PS is the bisector of ∠QPR of ΔPQR. Prove that .

#### Answer:

Let us draw a line segment RT parallel to SP which intersects extended line segment QP at point T.

Given that, PS is the angle bisector of ∠QPR.

∠QPS = ∠SPR … (1)

By construction,

∠SPR = ∠PRT (As PS || TR) … (2)

∠QPS = ∠QTR (As PS || TR) … (3)

Using these equations, we obtain

∠PRT = ∠QTR

∴ PT = PR

By construction,

PS || TR

By using basic proportionality theorem for ΔQTR,

$\frac{\mathrm{QS}}{\mathrm{SR}}=\frac{\mathrm{QP}}{\mathrm{PT}}$

##### Video Solution for triangles (Page: 152 , Q.No.: 1)

NCERT Solution for Class 10 math - triangles 152 , Question 1

#### Question 2:

In the given figure, D is a point on hypotenuse AC of ΔABC, DM ⊥ BC and DN ⊥ AB, Prove that:

(i) DM2 = DN.MC

(ii) DN2 = DM.AN

#### Answer:

(i)Let us join DB.

We have, DN || CB, DM || AB, and ∠B = 90°

∴ DMBN is a rectangle.

∴ DN = MB and DM = NB

The condition to be proved is the case when D is the foot of the perpendicular drawn from B to AC.

∴ ∠CDB = 90°

⇒ ∠2 + ∠3 = 90° … (1)

In ΔCDM,

∠1 + ∠2 + ∠DMC = 180°

⇒ ∠1 + ∠2 = 90° … (2)

In ΔDMB,

∠3 + ∠DMB + ∠4 = 180°

⇒ ∠3 + ∠4 = 90° … (3)

From equation (1) and (2), we obtain

∠1 = ∠3

From equation (1) and (3), we obtain

∠2 = ∠4

In ΔDCM and ΔBDM,

∠1 = ∠3 (Proved above)

∠2 = ∠4 (Proved above)

∴ ΔDCM ∼ ΔBDM (AA similarity criterion)

⇒ DM2 = DN × MC

(ii) In right triangle DBN,

∠5 + ∠7 = 90° … (4)

In right triangle DAN,

∠6 + ∠8 = 90° … (5)

D is the foot of the perpendicular drawn from B to AC.

∴ ∠ADB = 90°

⇒ ∠5 + ∠6 = 90° … (6)

From equation (4) and (6), we obtain

∠6 = ∠7

From equation (5) and (6), we obtain

∠8 = ∠5

In ΔDNA and ΔBND,

∠6 = ∠7 (Proved above)

∠8 = ∠5 (Proved above)

∴ ΔDNA ∼ ΔBND (AA similarity criterion)

⇒ DN2 = AN × NB

⇒ DN2 = AN × DM (As NB = DM)

##### Video Solution for triangles (Page: 152 , Q.No.: 2)

NCERT Solution for Class 10 math - triangles 152 , Question 2

#### Question 3:

In the given figure, ABC is a triangle in which ∠ABC> 90° and AD ⊥ CB produced. Prove that AC2 = AB2 + BC2 + 2BC.BD.

#### Answer:

Applying Pythagoras theorem in ΔADB, we obtain

AB2 = AD2 + DB2 … (1)

Applying Pythagoras theorem in ΔACD, we obtain

AC2 = AD2 + DC2

AC2 = AD2 + (DB + BC)2

AC2 = AD2 + DB2 + BC2 + 2DB × BC

AC2 = AB2 + BC2 + 2DB × BC [Using equation (1)]

##### Video Solution for triangles (Page: 152 , Q.No.: 3)

NCERT Solution for Class 10 math - triangles 152 , Question 3

#### Question 4:

In the given figure, ABC is a triangle in which ∠ABC < 90° and AD ⊥ BC. Prove that AC2 = AB2 + BC2 − 2BC.BD.

#### Answer:

Applying Pythagoras theorem in ΔADB, we obtain

AD2 + DB2 = AB2

⇒ AD2 = AB2 − DB2 … (1)

Applying Pythagoras theorem in ΔADC, we obtain

AD2 + DC2 = AC2

AB2 − BD2 + DC2 = AC2 [Using equation (1)]

AB2 − BD2 + (BC − BD)2 = AC2

AC2 = AB2 − BD2 + BC2 + BD2 −2BC × BD

= AB2 + BC2 − 2BC × BD

##### Video Solution for triangles (Page: 152 , Q.No.: 4)

NCERT Solution for Class 10 math - triangles 152 , Question 4

#### Question 5:

In the given figure, AD is a median of a triangle ABC and AM ⊥ BC. Prove that:

(i)

(ii)

(iii)

#### Answer:

(i) Applying Pythagoras theorem in ΔAMD, we obtain

AM2 + MD2 = AD2 … (1)

Applying Pythagoras theorem in ΔAMC, we obtain

AM2 + MC2 = AC2

AM2 + (MD + DC)2 = AC2

(AM2 + MD2) + DC2 + 2MD.DC = AC2

AD2 + DC2 + 2MD.DC = AC2 [Using equation (1)]

Using the result, , we obtain

(ii) Applying Pythagoras theorem in ΔABM, we obtain

AB2 = AM2 + MB2

= (AD2 − DM2) + MB2

= (AD2 − DM2) + (BD − MD)2

= AD2 − DM2 + BD2 + MD2 − 2BD × MD

= AD2 + BD2 − 2BD × MD

(iii)Applying Pythagoras theorem in ΔABM, we obtain

AM2 + MB2 = AB2 … (1)

Applying Pythagoras theorem in ΔAMC, we obtain

AM2 + MC2 = AC2 … (2)

Adding equations (1) and (2), we obtain

2AM2 + MB2 + MC2 = AB2 + AC2

2AM2 + (BD − DM)2 + (MD + DC)2 = AB2 + AC2

2AM2+BD2 + DM2 − 2BD.DM + MD2 + DC2 + 2MD.DC = AB2 + AC2

2AM2 + 2MD2 + BD2 + DC2 + 2MD (− BD + DC) = AB2 + AC2

##### Video Solution for triangles (Page: 152 , Q.No.: 5)

NCERT Solution for Class 10 math - triangles 152 , Question 5

#### Question 6:

Prove that the sum of the squares of the diagonals of parallelogram is equal to the sum of the squares of its sides.

#### Answer:

Let ABCD be a parallelogram.

Let us draw perpendicular DE on extended side AB, and AF on side DC.

Applying Pythagoras theorem in ΔDEA, we obtain

DE2 + EA2 = DA2 … (i)

Applying Pythagoras theorem in ΔDEB, we obtain

DE2 + EB2 = DB2

DE2 + (EA + AB)2 = DB2

(DE2 + EA2) + AB2 + 2EA × AB = DB2

DA2 + AB2 + 2EA × AB = DB2(ii)

Applying Pythagoras theorem in ΔADF, we obtain

AD2 = AF2 + FD2

Applying Pythagoras theorem in ΔAFC, we obtain

AC2 = AF2 + FC2

= AF2 + (DC − FD)2

= AF2 + DC2 + FD2 − 2DC × FD

= (AF2 + FD2) + DC2 − 2DC × FD

AC2 = AD2 + DC2 − 2DC × FD … (iii)

Since ABCD is a parallelogram,

AB = CD … (iv)

And, BC = AD … (v)

In ΔDEA and ΔADF,

∠DEA = ∠AFD (Both 90°)

∠EAD = ∠ADF (EA || DF)

AD = AD (Common)

∴ ΔEAD ΔFDA (AAS congruence criterion)

⇒ EA = DF … (vi)

Adding equations (i) and (iii), we obtain

DA2 + AB2 + 2EA × AB + AD2 + DC2 − 2DC × FD = DB2 + AC2

DA2 + AB2 + AD2 + DC2 + 2EA × AB − 2DC × FD = DB2 + AC2

BC2 + AB2 + AD2 + DC2 + 2EA × AB − 2AB × EA = DB2 + AC2

[Using equations (iv) and (vi)]

AB2 + BC2 + CD2 + DA2 = AC2 + BD2

##### Video Solution for triangles (Page: 153 , Q.No.: 6)

NCERT Solution for Class 10 math - triangles 153 , Question 6

#### Question 7:

In the given figure, two chords AB and CD intersect each other at the point P. prove that:

(i) ΔAPC ∼ ΔDPB

(ii) AP.BP = CP.DP

#### Answer:

Let us join CB.

(i) In ΔAPC and ΔDPB,

∠APC = ∠DPB (Vertically opposite angles)

∠CAP = ∠BDP (Angles in the same segment for chord CB)

ΔAPC ∼ ΔDPB (By AA similarity criterion)

(ii) We have already proved that

ΔAPC ∼ ΔDPB

We know that the corresponding sides of similar triangles are proportional.

∴ AP. PB = PC. DP

##### Video Solution for triangles (Page: 153 , Q.No.: 7)

NCERT Solution for Class 10 math - triangles 153 , Question 7

#### Question 8:

In the given figure, two chords AB and CD of a circle intersect each other at the point P (when produced) outside the circle. Prove that

(i) ΔPAC ∼ ΔPDB

(ii) PA.PB = PC.PD

#### Answer:

(i) In ΔPAC and ΔPDB,

∠P = ∠P (Common)

∠PAC = ∠PDB (Exterior angle of a cyclic quadrilateral is ∠PCA = ∠PBD equal to the opposite interior angle)

∴ ΔPAC ∼ ΔPDB

(ii)We know that the corresponding sides of similar triangles are proportional.

∴ PA.PB = PC.PD

##### Video Solution for triangles (Page: 153 , Q.No.: 8)

NCERT Solution for Class 10 math - triangles 153 , Question 8

#### Question 9:

In the given figure, D is a point on side BC of ΔABC such that. Prove that AD is the bisector of ∠BAC.

#### Answer:

Let us extend BA to P such that AP = AC. Join PC.

It is given that,

By using the converse of basic proportionality theorem, we obtain

AD || PC

⇒ ∠BAD = ∠APC (Corresponding angles) … (1)

And, ∠DAC = ∠ACP (Alternate interior angles) … (2)

By construction, we have

AP = AC

⇒ ∠APC = ∠ACP … (3)

On comparing equations (1), (2), and (3), we obtain

∠BAD = ∠APC

⇒ AD is the bisector of the angle BAC.

##### Video Solution for triangles (Page: 153 , Q.No.: 9)

NCERT Solution for Class 10 math - triangles 153 , Question 9

#### Question 10:

Nazima is fly fishing in a stream. The tip of her fishing rod is 1.8 m above the surface of the water and the fly at the end of the string rests on the water 3.6 m away and 2.4 m from a point directly under the tip of the rod. Assuming that her string (from the tip of her rod to the fly) is taut, ho much string does she have out (see Fig. 6.64)? If she pulls in the string at the rate of 5 cm per second, what will be the horizontal distance of the fly from her after 12 seconds?

#### Answer:

Let AB be the height of the tip of the fishing rod from the water surface. Let BC be the horizontal distance of the fly from the tip of the fishing rod.

Then, AC is the length of the string.

AC can be found by applying Pythagoras theorem in ΔABC.

AC2 = AB2 + BC2

AB2 = (1.8 m)2 + (2.4 m)2

AB2 = (3.24 + 5.76) m2

AB2 = 9.00 m2

Thus, the length of the string out is 3 m.

She pulls the string at the rate of 5 cm per second.

Therefore, string pulled in 12 seconds = 12 × 5 = 60 cm = 0.6 m

Let the fly be at point D after 12 seconds.

Length of string out after 12 seconds is AD.

AD = AC − String pulled by Nazima in 12 seconds

= (3.00 − 0.6) m

= 2.4 m

In ΔADB,

AB2 + BD2 = AD

(1.8 m)2 + BD2 = (2.4 m)2

BD2 = (5.76 − 3.24) m2 = 2.52 m2

BD = 1.587 m

Horizontal distance of fly = BD + 1.2 m

= (1.587 + 1.2) m

= 2.787 m

= 2.79 m

##### Video Solution for triangles (Page: 153 , Q.No.: 10)

NCERT Solution for Class 10 math - triangles 153 , Question 10

View NCERT Solutions for all chapters of Class 10