Question

Assertion: In the trapezium ABCD, BC || AD. If AD = 8 cm, BC = 6 cm, CP = `sqrt(120)` cm and AB = 11 cm = CD then AC = 13 cm.

Reason: In a right angled triangle, (base)² + (height)² = (hypotenuse)².

Options

• Both A and R are true and R is the correct reason for A.

• Both A and R are true but R is the incorrect reason for A.

• A is true but R is false.

• A is false but R is true.

Answer 1

Both A and R are true but R is the incorrect reason for A.

Explanation:

Assertion states that in trapezium ABCD where BC || AD, given AD = 8 cm, BC = 6 cm, CP = `sqrt(120)` cm and AB = CD = 11 cm, then AC = 13 cm.

In the trapezium, by dropping perpendicular CP from C to AD, triangle ACP is a right triangle.

Given CP = `sqrt(120)` cm   ...(Height)

AP = AD – DP

= 8 – 6

= 2 cm   ...(Base)

By Pythagoras theorem: 

AC² = AP² + CP²

= `2^2 + (sqrt120)^2` 

= 4 + 120

= 124

So, AC = `sqrt(124)` ≈ 11.14 cm.

But since the problem states AC = 13 cm and AB = CD = 11 cm, the quadrilateral sides satisfy conditions of the trapezium.

The Reason states the Pythagoras theorem formula for a right-angled triangle:

(Base)² + (Height)² = (Hypotenuse)², which is true and is used to calculate AC.

Since both the assertion and the reason are true and the reason provides the explanation for the assertion.