CAT 2024 Slot 2 QA Question 19

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

The coordinates of the three vertices of a triangle are:

(1, 2) , (7, 2) , and

(1, 10)$. Then the radius of the in circle of the triangle 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

The radius of the in-circle (inradius) of a triangle is given by the formula: $

r = \frac{A}{s}

where

A

is the area of the triangle and

s

is the semi-perimeter.

🔢 Step-by-Step Solution

Identify the Triangle Type and Calculate Sides:

The triangle has vertices at

(1, 2) , (7, 2) , and

(1, 10)$.

Calculate side lengths using the distance formula:

AB = \sqrt{(7-1)^2 + (2-2)^2} = 6

AC = \sqrt{(1-1)^2 + (10-2)^2} = 8

BC = \sqrt{(7-1)^2 + (2-10)^2} = 10

The triangle is right-angled with legs 6 and 8, and hypotenuse 10.

Calculate the Area (A):

For a right-angled triangle, area

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

A = \frac{1}{2} \times 6 \times 8 = 24

Calculate the Semi-Perimeter (s):

s = \frac{a + b + c}{2} = \frac{6 + 8 + 10}{2} = 12

Compute the Inradius (r):

r = \frac{A}{s} = \frac{24}{12} = 2

⚡ 30-Second Shortcut

For right-angled triangles, the inradius can be quickly calculated using:

r = \frac{a + b - c}{2} where c is the hypotenuse. Plugging in: r = \frac{6 + 8 - 10}{2} = \frac{4}{2} = 2

🎯 Final Answer

Correct Answer: 2

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