CAT 2023 Slot 1 QA Question 19

Type-in-the-answer (no negative marking) · Geometry · Triangles · Try it, then check the answer and solution below.

In a right-angled triangle

\triangle

A B C , the altitude

A B

5 \mathrm{~cm} , and the base

B C

is

12 \mathrm{~cm}

.

and

are two points on

B C

such that the areas of

\triangle \mathrm{ABP}, \triangle \mathrm{ABQ}

and

\triangle \mathrm{ABC}

are in arithmetic progression. If the area of

\triangle \mathrm{ABC}

is

1.5

times the area of

\triangle \mathrm{ABP} , the length of PQ in cm , is

TITA Answer:

⌨️ Shortcuts:[1-4 / A-D] Select[Enter] Check[B] Star[S] Solution

Official Convenor Answer Key: 2

Answer and solution

Answer: 2

📌 Core Concept

In a right-angled triangle, the area can be calculated using the formula:

\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}

🔢 Step-by-Step Solution

Identify the triangle and given values:

Triangle

\triangle ABC

is right-angled at

B

AB = 5 \, \text{\text{ cm}}

(altitude),

BC = 12 \, \text{\text{ cm}}

(base).

The area of

\triangle ABC

is:

\text{Area}_{ABC} = \frac{1}{2} \times AB \times BC = \frac{1}{2} \times 5 \times 12 = 30 \, \text{\text{ cm}}^2

Set up the areas in arithmetic progression:

Let the areas of

\triangle ABP , \triangle ABQ , and

\triangle ABC

be

A_1 , A_2 , and

A_3

respectively.

Given

A_3 = 1.5 \times A_1

and

A_1, A_2, A_3

are in AP.

Therefore,

2A_2 = A_1 + A_3

and substituting \( A

Keep going