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#### Question 1:

Draw a circle of radius 6 cm. From a point 10 cm away from its centre, construct the pair of tangents to the circle and measure their lengths.

Given that

Construct a circle of radius, and form its centre, construct the pair of tangents to the circle.

Find the length of tangents.

We follow the following steps to construct the given

Step of construction

Step: I- First of all we draw a circle of radius.

Step: II- Make a point P at a distance of, and join .

Step: III -Draw a right bisector of, intersecting at Q .

Step: IV- Taking Q as centre and radius, draw a circle to intersect the given circle at T and T’.

Step: V- Joins PT and PT’ to obtain the require tangents.

Thus, are the required tangents.

Find the length of tangents.

As we know that and is right triangle.

Therefore,

In,

Thus, the length of tangents

#### Question 2:

Draw a circle of radius 3 cm. Take two points P and Q on one of its extended diameter each at a distance of 7 cm from its centre. Draw tangents to the circle from these two points P and Q.

Given that

Construct a circle of radius, and extended diameter each at distance of 7cm from its centre. Construct the pair of tangents to the circle from these two points .

We follow the following steps to construct the given

Step of construction

Step: I- First of all we draw a circle of radius.

Step: II- Make a line CD .

Step: III-Extend the line CD in such a way that point

Step: IV- CP at a distance of, and join draw a right bisector of, intersecting at R.

Step V:- Similarly, DQ at a distance of, and join draw a right bisector of, intersecting at S.

Step VI: Taking R and S as centre and radius, draw a circle to intersect the given circle at T and T’

B and B ’respectively.

Step: VII- Joins PT and PT’ as well as QB and QB’ to obtain the require tangents.

Thus, are the required tangents.

#### Question 4:

Draw a pair of tangents to a circle of radius 4.5 cm, which are inclined to each other at an angle of 45°.                         [CBSE 2013]

Steps of Construction

Step 1. Draw a circle with centre O and radius 4.5 cm.

Step 2. Draw any diameter AOB of the circle.

Step 3. Construct $\angle$BOC = 45º such that radius OC cuts the circle at C.

Step 4. Draw AM ⊥ AB and CN ⊥ OC. Suppose AM and CN intersect each other at P.

Here, AP and CP are the pair of tangents to the circle inclined to each other at an angle of 45º.

#### Question 5:

Draw two tangents to a circle of radius 3.5 cm from a point P at a distance of 6.2 from its centre.                          [CBSE 2013]

Steps of Construction

Step 1. Draw a circle with O as centre and radius 3.5 cm.

Step 2. Mark a point P outside the circle such that OP = 6.2 cm.

Step 3. Join OP. Draw the perpendicular bisector XY of OP, cutting OP at Q.

Step 4. Draw a circle with Q as centre and radius PQ (or OQ), to intersect the given circle at the points T and T'.

Step 5. Join PT and PT'.

Here, PT and PT' are the required tangents.

#### Question 3:

Draw a line segment AB of length 8 cm. Taking A as centre, draw a circle of radius 4 cm and taking B as centre, draw another circle of radius 3 cm. Construct tangents to each circle from the centre of the other circle.

Given that

Construct a circle of radius, and extended diameter each at distance of 7cm from its centre. Construct the pair of tangents to the circle from these two points .

We follow the following steps to construct the given

Step of construction

Step: I First of all we draw a line.

Step: II taking A as a centre and draw a circle of radius. Similarly, taking B as a centre and draw a circle of radius.

Step: III draw the perpendicular bisector of

Step IV : draw the another circle with taking the bisector point as centre and radius = mid point of which cut the point

Step: V joins and respectively. as well as to obtain the require tangents.

Thus, are the required tangents.

#### Question 1:

Determine a point which divides a line segment of length 12 cm internally in the ratio 2 : 3 Also, justify your construction.

Given that

Determine a point which divides a line segment of lengthinternally in the ratio of.

We follow the following steps to construct the given

Step of construction

Step: I- First of all we draw a line segment.

Step: II- We draw a ray making an acute anglewith.

Step: III- Draw a ray parallel to AX by making an acute angle.

Step IV- Mark of two points on and three points on in such a way that.

Step: V- Joins and this line intersects at a point P.

Thus, P is the point dividing internally in the ratio of

Justification:

In we have

And

So, AA similarity criterion, we have

#### Question 2:

Divide a line segment of length 9 cm internally in the ratio 4 : 3. Also, give justification of the construction.

Given that

Determine a point which divides a line segment of lengthinternally in the ratio of.

We follow the following steps to construct the given

Step of construction

Step: I- First of all we draw a line segment.

Step: II- We draw a ray making an acute anglewith.

Step: III- Draw a ray parallel to AX by making an acute angle.

Step IV- Mark of two points on and three points on in such a way that.

Step: V- Joins and this line intersects at a point P.

Thus, P is the point dividing internally in the ratio of

Justification:

In we have

And

So, AA similarity criterion, we have

#### Question 3:

Divide a line segment of length 14 cm internally in the ratio 2 : 5. Also, justify your construction.

Given that

Determine a point which divides a line segment of lengthinternally in the ratio of.

We follow the following steps to construct the given

Step of construction

Step: I-First of all we draw a line segment.

Step: II- We draw a ray making an acute anglewith.

Step: III- Draw a ray parallel to AX by making an acute angle.

Step IV- Mark of two points on and three points on in such a way that.

Step: V- Joins and this line intersects at a point P.

Thus, P is the point dividing internally in the ratio of

Justification:

In we have

And

So, AA similarity criterion, we have

#### Question 1:

Construct a triangle of sides 4 cm, 5 cm and 6 cm and then a triangle similar to it whose sides are (2/3)  of the corresponding sides of it.

Given that

Construct a triangle of sides and then a triangle similar to it whose sides are of the corresponding sides of it.

We follow the following steps to construct the given

Step of construction

Step: I- First of all we draw a line segment.

Step: II- With A as centre and radius, draw an arc.

Step: III- With B as centre and radius, draw an arc, intersecting the arc drawn in step II at C.

Step: IV- Joins AC and BC to obtain.

Step: V- Below AB, makes an acute angle.

Step: VI- Along AX, mark off three points such that

Step: VII- Join.

Step: VIII- Since we have to construct a triangle each of whose sides is two-third of the corresponding sides of.

So, we take two parts out of three equal parts on AX from point draw and meeting AB at C’.

Step: IX- From B’ draw and meeting AC at C’

Thus, is the required triangle, each of whose sides is two third of the corresponding sides of.

#### Question 2:

Construct a triangle similar to a given ΔABC such that each of its sides is (5/7)th  of the corresponding sides of Δ ABC. It is given that AB - 5 cm, BC = 7 cm and ∠ABC = 50°.

Given that

Construct a triangle similar to a triangle ABC such that each of sides is of the corresponding sides of triangle ABC.

We follow the following steps to construct the given

Step of construction

Step: I- First of all we draw a line segment.

Step: II- With B as centre and draw an angle.

Step: III- With B as centre and radius, draw an arc, cut the line BY drawn in step II at C.

Step: IV- Joins AC to obtain.

Step: V- Below AB, makes an acute angle.

Step: VI- Along AX, mark off seven points such that

Step: VII-Join.

Step: VIII- Since we have to construct a triangle each of whose sides is of the corresponding sides of.

So, we take five parts out of seven equal parts on AX from point draw and meeting AB at B’.

Step: IX- From B’ draw and meeting AC at C’

Thus, is the required triangle, each of whose sides is of the corresponding sides of.

#### Question 3:

Construct a triangle similar to a given ΔABC such that each of its sides is (2/3)rd of the corresponding sides of ΔABC. It is given that BC = 6 cm, ∠B = 50° and ∠C = 60°.

Given that

Construct a triangle of given data, and then a triangle similar to it whose sides are of the corresponding sides of .

We follow the following steps to construct the given

Step of construction

Step: I- First of all we draw a line segment.

Step: II- With B as centre draw an angle.

Step: III- With C as centre draw an anglewhich intersecting the line drawn in step II at A.

Step: IV- Joins AB and AC to obtain.

Step: V -Below BC, makes an acute angle.

Step: VI -Along BX, mark off three points such that

Step: VII -Join.

Step: VIII -Since we have to construct a triangle each of whose sides is two-third of the corresponding sides of.

So, we take two parts out of three equal parts on BX from point draw and meeting BC at C’.

Step: IX -From C’ draw and meeting AB at A

Thus, is the required triangle, each of whose sides is two third of the corresponding sides of.

#### Question 4:

Draw a ΔABC in which BC = 6 cm, AB = 4 cm and AC = 5 cm. Draw a triangle similar to ΔABC with its sides equal to (3/4)th of the corresponding sides of ΔABC.

Given that

Construct a triangle of sides and then a triangle similar to it whose sides are of the corresponding sides of.

We follow the following steps to construct the given

Step of construction

Step: I- First of all we draw a line segment.

Step: II- With A as centre and radius, draw an arc.

Step: III -With B as centre and radius, draw an arc, intersecting the arc drawn in step II at C.

Step: IV -Joins AC and BC to obtain.

Step: V -Below AB, makes an acute angle.

Step: VI -Along AX, mark off four points such that

Step: VII -Join.

Step: VIII -Since we have to construct a triangle each of whose sides is of the corresponding sides of.

So, we take three parts out of four equal parts on AX from point draw and meeting AB at B’.

Step: IX- From B’ draw and meeting AC at C’

Thus, is the required triangle, each of whose sides is of the corresponding sides of.

#### Question 5:

Construct a triangle with sides 5 cm, 6 cm and 7 cm and then another triangle whose sides are 7/5 of the corresponding sides of the first triangle.

Given that

Construct a triangle of sides and then a triangle similar to it whose sides are of the corresponding sides of.

We follow the following steps to construct the given

Step of construction

Step: I- First of all we draw a line segment.

Step: II- With A as centre and radius, draw an arc.

Step: III- With B as centre and radius, draw an arc, intersecting the arc drawn in step II at C.

Step: IV- Joins AC and BC to obtain.

Step: V- Below AB, makes an acute angle.

Step: VI- Along AX, mark off seven points such that

Step: VII-Join.

Step: VIII- Since we have to construct a triangle each of whose sides is of the corresponding sides of.

So, we draw a line on AX from point which is and meeting AB at B’.

Step: IX- From B’ point draw and meeting AC at C’

Thus, is the required triangle, each of whose sides is of the corresponding sides of.

#### Question 6:

Draw a right triangle ABC in which AC = AB = 4.5 cm and ∠A = 90°. Draw a triangle similar to ΔABC with its sides equal to (5/4)th of the corresponding sides of ΔABC.

Given that

Construct a right triangle of sides and then a triangle similar to it whose sides are of the corresponding sides of.

We follow the following steps to construct the given

Step of construction

Step: I- First of all we draw a line segment.

Step: II- With A as centre and draw an angle.

Step: III- With A as centre and radius.

Step: IV- Join BC to obtain.

Step: V- Below AB, makes an acute angle.

Step: VI- Along AX, mark off five points such that

Step: VII-Join.

Step: VIII- Since we have to construct a triangle each of whose sides is of the corresponding sides of.

So, we draw a line on AX from point which is and meeting AB at B’.

Step: IX- From B’ point draw and meeting AC at C’

Thus, is the required triangle, each of whose sides is of the corresponding sides of.

#### Question 7:

Draw a right triangle in which the sides (other than hypotenuse) are of lengths 5 cm and 4 cm. Then construct another triangle whose sides are 5/3 times the corresponding sides of the given triangle.

Given that

Construct a right triangle of sides and then a triangle similar to it whose sides are of the corresponding sides of.

We follow the following steps to construct the given

Step of construction

Step: I- First of all we draw a line segment.

Step: II- With A as centre and draw an angle.

Step: III- With A as centre and radius.

Step: IV -Join BC to obtain.

Step: V -Below AB, makes an acute angle.

Step: VI -Along AX, mark off five points such that

Step: VII -Join.

Step: VIII -Since we have to construct a triangle each of whose sides is of the corresponding sides of.

So, we draw a line on AX from point which is and meeting AB at B’.

Step: IX -From B’ point draw and meeting AC at C’

Thus, is the required triangle, each of whose sides is of the corresponding sides of.

#### Question 8:

Construct a ΔABC in which AB = 5 cm. ∠B = 60° altitude CD = cm. Construct a ΔAQR similar to ΔABC such that side ΔAQR is 1.5 times that of the corresponding sides of ΔACB

Given that

Construct a trianglein which and then a triangle similar to it whose sides are of the corresponding sides of.

We follow the following steps to construct the given

Step of construction

Step: I- First of all we draw a line segment.

Step: II- With B as centre and draw an angle.

Step: III -From point A and B construct which cut the line BS at point C

Step: IV- Join AC to obtain.

Step: V- Below AB, makes an acute angle.

Step: VI- Along AX, mark off five points such that

Step: VII -Join.

Step: VIII -Since we have to construct a triangle each of whose sides is of the corresponding sides of.

So, we draw a line on AX from point which is and meeting AB at Q.

Step: IX- From Q point draw and meeting AC at R

Thus, is the required triangle, each of whose sides is of the corresponding sides of.

#### Question 9:

Construct an isosceles triangle whose base is 8 cm and altitude 4 cm and then another triangle whose sides are 3/2 times the corresponding sides of the isosceles triangle.

Given that

Construct an isosceles triangle ABC in which AB = BC = 6 cm and altitude = 4 cm then another triangle similar to it whose sides are $\frac{3}{2}$ of the corresponding sides of$△ABC$.

We follow the following steps to construct the given

Step of construction

Step: I- First of all we draw a line segment AB = 6 cm.

Step: II- With B as centre and radius = BC = 6 cm, draw an arc.

Step: III- From point A and B construct which cut the line BS at point C

Step: IV -Join AC to obtain.

Step: V- Below AB, makes an acute angle.

Step: VI -Along AX, mark off five points such that

Step: VII- Join.

Step: VIII -Since we have to construct a triangle each of whose sides is of the corresponding sides of.

So, we draw a line on AX from point which is and meeting AB at Q.

Step: IX -From Q point draw and meeting AC at R

Thus, is the required triangle, each of whose sides is of the corresponding sides of.

#### Question 10:

Draw a ΔABC with side BC = 6 cm. AB = 5 cm and ∠ ABC = 60°. Then, construct a triangle whose sides are (3/4)th of the corresponding sides of the ΔABC.

Given that

Construct a of given data, and then a triangle similar to it whose sides are of the corresponding sides of.

We follow the following steps to construct the given

Step of construction

Step: I- First of all we draw a line segment.

Step: II- With B as centre draw an angle.

Step: III- With B as centre and radius, draw an arc.

Step: IV- Join AC to obtain.

Step: V -Below AB, makes an acute angle.

Step: VI -Along AX, mark off four points such that

Step: VII -Join.

Step: VIII -Since we have to construct a triangle each of whose sides is of the corresponding sides of.

So, we take three parts out of four equal parts on AX from point draw and meeting AB at B’.

Step: IX- From B’ draw and meeting AC at C’

Thus, is the required triangle, each of whose sides is of the corresponding sides of.

#### Question 11:

Construct a triangle similar to Δ ABC in which AB = 4.6 cm, BC = 5.1 cm, ∠A = 60° with scale factor 4 : 5.

Given that

Construct a of given data, and then a triangle similar to it whose sides are of the corresponding sides of.

We follow the following steps to construct the given

Step of construction

Step: I- First of all we draw a line segment.

Step: II- With A as centre draw an angle.

Step: III- With B as centre and radius, draw an arc, intersecting the arc drawn in step II at C.

Step: IV- Joins BC to obtain.

Step: V- Below AB, makes an acute angle.

Step: VI- Along AX, mark off five points such that

Step: VII- Join.

Step: VIII- Since we have to construct a triangle each of whose sides is of the corresponding sides of.

So, we take four parts out of five equal parts on AX from point draw and meeting AB at B’.

Step: IX- From B’ draw and meeting AC at C’

Thus, is the required triangle, each of whose sides is of the corresponding sides of.

#### Question 12:

Construct a triangle similar to a given ΔXYZ with its sides equal to (3/4)th of the corresponding sides of ΔXYZ. Write the steps of construction.

Given that

Construct a of given data, and then a triangle similar to it whose sides are of the corresponding sides of.

We follow the following steps to construct the given

Step of construction

Step: I- First of all we draw a line segment.

Step: II- With Y as centre draw an angle.

Step: III- With Y as centre and radius, draw an arc.

Step: IV- Join XZ to obtain.

Step: V- Below XY, makes an acute angle.

Step: VI -Along XP, mark off four points such that

Step: VII- Join.

Step: VIII- Since we have to construct a triangle each of whose sides is of the corresponding sides of.

So, we take three parts out of four equal parts on XP from point draw and meeting XY at Y’.

Step: IX- From Y’ draw and meeting XZ at Z

Thus, is the required triangle, each of whose sides is of the corresponding sides of.

#### Question 13:

Draw a right triangle in which sides (other than the hypotenuse) are of lengths 8 cm and 6 cm. Then construct another triangle whose sides are $\frac{3}{4}$ times the corresponding sides of the first triangle.

Given that

Construct a right triangle of sides and then a triangle similar to it whose sides are of the corresponding sides of.

We follow the following steps to construct the given

Step of construction

Step: I- First of all we draw a line segment.

Step: II- With A as centre and draw an angle.

Step: III- With A as centre and radius.

Step: IV-Join BC to obtain right .

Step: V- Below AB, makes an acute angle.

Step: VI- Along AX, mark off five points such that

Step: VII- Join.

Step: VIII -Since we have to construct a triangle each of whose sides is of the corresponding sides of right .

So, we draw a line on AX from point which is and meeting AB at B’.

Step: IX- From B’ point draw and meeting AC at C’

Thus, is the required triangle, each of whose sides is of the corresponding sides of.

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