Pythagoras Theorem — Assertion-Reason Type Questions

Assertion (A): If the sides of a rectangular plot are 80 m and 60 m, then the length of its diagonal is 100 m.

Reason (R): In a rectangle, all angles are 90°.

1. Assertion (A) is true, Reason (R) is false.

2. Assertion (A) is false, Reason (R) is true.

3. Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct reason for Assertion (A).

4. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct reason (or explanation) for Assertion (A).

Answer

A rectangle is a quadrilateral with four right angles.

Let ABCD be the rectangular plot.

Thus, in a rectangle, all angles are 90°.

∴ Reason (R) is true.

Given:

Side 1 (AB) = 80 m

Side 2 (BC) = 60 m

Let d be the length of the diagonal (AC). Using pythagoras theorem in triangle ABC,

⇒ AC² = AB² + BC²

⇒ d² = 80² + 60²

⇒ d² = 6400 + 3600

⇒ d² = 10000

⇒ d = $\sqrt{10000}$

⇒ d = ± 100 m

Since, length cannot be negative, so, diagonal = 100 m.

∴ Assertion (A) is true.

∴ Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct reason (or explanation) for Assertion (A).

Hence, option 3 is the correct option.

Assertion (A): If triangle ABC is isosceles with AC = BC and AB² = 2AC², then triangle ABC is right angled.

Reason (R): If in a triangle ABC, we have AB² = 2AC² then we can conclude that it is right angled triangle.

1. Assertion (A) is true, Reason (R) is false.

2. Assertion (A) is false, Reason (R) is true.

3. Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct reason for Assertion (A).

4. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct reason (or explanation) for Assertion (A).

Answer

Given, triangle ABC is isosceles with AC = BC and, AB² = 2AC²

⇒ AB² = AC² + AC²

⇒ AB² = AC² + BC²

This equation AB² = AC² + BC² is the form of the Pythagorean Theorem where AB is the hypotenuse and AC and BC are the other sides of te triangle.

Since, pythagoras theorem is satisfied, thus ABC is a right angle triangle.

∴ Assertion (A) is true.

Alone with the relation, AB² = 2AC², we cannot conclude that it is right angled triangle because there is no mention of the third side BC.

∴ Reason (R) is false.

∴ Assertion (A) is true, Reason (R) is false.

Hence, option 1 is the correct option.

Assertion (A): A triangle with sides 2 cm, 3 cm, 4 cm is not a right angled triangle.

Reason (R): A triangle with sides 2 cm, 3 cm, 4 cm is a scalene triangle.

1. Assertion (A) is true, Reason (R) is false.

2. Assertion (A) is false, Reason (R) is true.

3. Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct reason for Assertion (A).

4. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct reason (or explanation) for Assertion (A).

Answer

Given the sides: a = 2 cm, b = 3 cm, c = 4 cm.

The longest side is c = 4 cm.

Let's check if c² = a² + b²

Taking L.H.S.,

c² = 4² = 16.

Taking R.H.S.,

a² + b² = 2² + 3²

= 4 + 9

= 13.

Since 16 ≠ 13, the condition for a right-angled triangle is not met.

Therefore, the triangle with sides 2 cm, 3 cm, 4 cm is not a right-angled triangle.

∴ Assertion (A) is true.

A scalene triangle is defined as a triangle in which all three sides have different lengths.

Given the side lengths are 2 cm, 3 cm, and 4 cm. All three lengths are distinct.

∴ Reason (R) is true.

∴ Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct reason (or explanation) for Assertion (A).

Hence, option 4 is the correct option.