R.d Sharma 2022 Mcqs Solutions for Class 9 Maths Chapter 16 - Construction

Answer:

The angles which can be constructed by bisecting the standard angles 30°, 45°, 60°, 90° or 180° are the angles that are possible to be constructed using just a compass and a ruler, i.e., multiples of 15°.

Here,
35° × 2 = 70°

40° × 2 = 80°

$47\frac{1°}{2}$ × 2 = 95° 

 $37\frac{1°}{2}$ × 2 = 75°

Since an angle of 75° can be constructed by drawing an angle bisector of the angles measuring 90° and 60°, and then its angle bisector measuring $37\frac{1°}{2}$can be constructed by drawing an angle bisector of the angles measuring 75° and 0° with the help of a ruler and compasses.

Hence, the correct answer is option (c).

Answer:

The angles which cannot be constructed by bisecting the standard angles 30°, 45°, 60°, 90° or 180° are the angles which are not possible to be constructing using just a compass and a ruler, i.e., multiples of 15°.

Here,
 $37\frac{1°}{2}$ × 2 = 75°

40° × 2 = 80°

$22\frac{1°}{2}$ × 2 = 45° 

 $67\frac{1°}{2}$ × 2 = 135°

Since the angle of 80°is not possible to be constructed with the help of a ruler and compasses.
As a result, constructing a 40° angle is impossible.

Hence, the correct answer is option (b).

Answer:

A triangle cannot be constructed if the sum of two sides is less than or equal to the third side.

AB + BC < AC

It can be written as

4 + BC < AC

4 < AC − BC

AC − BC > 4

Therefore, construction is not possible when the difference of BC and AC is equal to 4.5 cm.

Hence, the correct answer is option (b).

Answer:

A triangle cannot be constructed if the sum of two sides is less than or equal to the third side.

AB + BC < AC

It can be written as

AB + 6 < AC

6 < AC − AB

AC − AB > 6

Therefore, construction is not possible when the difference of BC and AC is equal to 6.9 cm.

Hence, the correct answer is option (a).

Answer:

A triangle can be constructed if the sum of two sides is more than or equal to the third side.

AB + BC > AC

It can be written as

AB + 3 > AC

3 > AC − AB

Therefore, construction is possible when the difference of BC and AC is equal to 2.8 cm.

Hence, the correct answer is option (d).

Answer:

A triangle can be constructed if the sum of two sides is more than or equal to the third side.

BC + CA > AB

It can be written as

BC + CA > 5

Therefore, construction is possible when BC + CA is equal to 8 cm.

Hence, the correct answer is option (c).

Answer:

Statement-2 (Reason): With the help of a ruler and a compass an angle of 135° can be constructed and 67.5° is the bisector of angle of 135°.

The angles which can be constructed by bisecting the standard angles 30°, 45°, 60°, 90° or 180° are the angles which are possible to be constructed using just a compass and a ruler, i.e., multiples of 15°.

​Here, 135° is a multiple of 15°. So,  135° can be constructed with the help of a ruler and a compass by bisecting angles 90° and 180°,i.e., $\frac{90°+180°}{2}=\frac{270°}{2}=135°$
Now, $\frac{135°}{2}=67.5°$
So,  67.5° can be constructed with the help of a ruler and a compass by bisecting the angle of 135°.

Thus, Statement-2 is true.

Statement-1 (Assertion): With the help of a ruler and a compass, it is possible to construct an angle of 67.5°.
Now, according to Statement-2, 67.5° is the bisector of angle of 135°and with the help of a ruler and a compass an angle of 135° can be constructed.

Thus, Statement-1 is true.
So, Statement-2 is true and Statement-2 is a correct explanation for Statement-1.

Hence, the correct answer is option (a).

Answer:

Statement-2 (Reason): With the help of a ruler and a compass an angle of 210° = 180° + 30° can be constructed and 105° is its bisector whose bisector is an angle of 52.5°.

The angles which can be constructed by bisecting the standard angles 30°, 45°, 60°, 90° or 180° are the angles that are possible to be constructed using just a compass and a ruler, i.e., multiples of 15°.

So, 210° = 180° + 30° can be constructed with the help of a ruler and a compass.
Now, $\frac{210°}{2}=105°$
So, 105° can be constructed with the help of a ruler and a compass by bisecting the angle of 210°.

Similarly, $\frac{105°}{2}=52.5°$
So, 52.5° can be constructed with the help of a ruler and a compass by bisecting the angle of 105°.

Thus, Statement-2 is true.

Statement-1 (Assertion): With the help of a ruler and a compass an angle of 52.5° can be constructed.
Now, according to Statement-2, 52.5° can be constructed with the help of a ruler and a compass by bisecting the angle of 105°.

Thus, Statement-1 is true.
So, Statement-2 is true and Statement-2 is a correct explanation for Statement-1.

Hence, the correct answer is option (a).

Answer:

Statement-2 (Reason): A triangle ABC can be constructed in which BC = 6 cm, ∠C = 30° and AC – AB = 4 cm.

A triangle can be constructed if the sum of two sides is more than or equal to the third side.

BC + AB > AC

It can be written as

6 + AB > AC

6 > AC − AB

Therefore, construction is possible when AC − AB is equal to 4 cm.

Thus, Statement-2 is true.

Statement-1 (Assertion): A triangle ABC can not be constructed in which AB = 4 cm, ∠A = 60° and BC – AC = 4.5 cm.

A triangle cannot be constructed if the sum of two sides is less than or equal to the third side.

AB + BC < AC

It can be written as

4 + AC < BC

4 < BC − AC

BC − AC > 4

Therefore, construction is not possible when the difference o BC and AC is equal to 4.5 cm.

Thus, Statement-1 is true.
So, Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.

Hence, the correct answer is option (b).

Answer:

Statement-2 (Reason): A triangle ABC can be constructed in which ∠B = 105°, ∠C = 90° and AB + BC + AC = 10 cm.

In triangle ABC, given that

∠B = 105°

∠C = 90°

AB + BC + AC = 10 cm

From the angle sum property of a triangle

∠A + ∠B + ∠C = 180°

Substituting the values

∠B + ∠C = 105° + 90° = 195°

So, it is not possible to construct a triangle ABC in which ∠B = 105°, ∠C = 90° and AB + BC + AC = 10 cm.

Thus, Statement-2 is false.

Statement-1 (Assertion): A triangle ABC can be constructed in which ∠B = 60°, ∠C = 45° and AB + BC + AC = 12 cm.

​In triangle ABC, given that

∠B = 60°

∠C = 45°

AB + BC + AC = 12 cm

From the angle sum property of a triangle

∠A + ∠B + ∠C = 180°

Substituting the values

∠B + ∠C = 60° + 45° = 105°

Here, 105° < 180° ⇒ ∠A = 75° with which it is possible to construct a triangle ABC in which ∠B = 60°, ∠C = 45° and AB + BC + AC = 12 cm.

Thus, Statement-1 is true.
So, Statement-1 is true, Statement-2 is false.

Hence, the correct answer is option (c).