Pythagoras Theorem — Assertion Reason Type Questions

Assertion (A): A man goes 15 m due West and then 20 m due North, then he is 25 m from the starting point.

Reason (R): According to Pythagoras theorem, in a triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

1. A is true, R is false.

2. A is false, R is true.

3. Both A and R are true.

4. Both A and R are false.

Answer

West and North are perpendicular directions, so the distance from the starting point is the hypotenuse of a right-angled triangle with legs 15 m and 20 m.

⇒ Distance² = 15² + 20²

⇒ Distance² = 225 + 400

⇒ Distance² = 625

⇒ Distance = $\sqrt{625}$ = 25 m

So the man is 25 m from the starting point.

So, Assertion (A) is true.

According to Pythagoras theorem, in a triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

So, Reason (R) is also true.

Hence, option 3 is the correct option.

Assertion (A): Angles B and C of ∆ABC are 69° and 21°, then AC² - AB² = BC².

Reason (R): In a triangle if sum of any two angles is 90° then the 3^rd angle is right angle.

1. A is true, R is false.

2. A is false, R is true.

3. Both A and R are true.

4. Both A and R are false.

Answer

Angle B + Angle C = 69° + 21° = 90°, so Angle A = 180° - 90° = 90°.

Since the right angle is at A, the hypotenuse is BC and by Pythagoras theorem BC² = AB² + AC².

The Assertion states AC² - AB² = BC², which is incorrect (the correct relation is AC² + AB² = BC²). So, Assertion (A) is false.

If the sum of any two angles is 90°, the third angle = 180° - 90° = 90°, a right angle. So, Reason (R) is true.

Hence, option 2 is the correct option.

Assertion (A): If you walk diagonally across a rectangular field 60 m long and 80 m wide then you will walk 40 m shorter.

Reason (R): Line joining any two non-consecutive vertices of a rectangle is a diagonal of the rectangle.

1. A is true, R is false.

2. A is false, R is true.

3. Both A and R are true.

4. Both A and R are false.

Answer

Length of the diagonal :

⇒ Diagonal² = 60² + 80²

⇒ Diagonal² = 3600 + 6400

⇒ Diagonal² = 10000

⇒ Diagonal = $\sqrt{10000}$ = 100 m

Walking along the two sides = 60 + 80 = 140 m

Distance saved = 140 - 100 = 40 m

So walking diagonally is 40 m shorter. So, Assertion (A) is true.

The line joining any two non-consecutive (opposite) vertices of a rectangle is a diagonal.

So, Reason (R) is also true.

Hence, option 3 is the correct option.