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CAT 2024 Slot 2 — QA Question 10

TrianglesEasy

The coordinates of the three vertices of a triangle are: (1, 2), (7, 2), and (1, 10). Then the radius of the incircle of the triangle is

Answer & solution

Answer: 2

Solution

Easy

The two given legs are horizontal and vertical, so this is a right triangle. For a right triangle the inradius is r=leg1+leg2−hypotenuse2r=\dfrac{\text{leg}_1+\text{leg}_2-\text{hypotenuse}}{2}.

(1,2) (7,2) (1,10) 6 8
1

Find the side lengths. (1,2)→(7,2)(1,2)\to(7,2) is horizontal, (1,2)→(1,10)(1,2)\to(1,10) is vertical, meeting at a right angle.

base=7−1=6,height=10−2=8⇒ hypotenuse=62+82=10\begin{aligned} &\text{base}=7-1=6,\qquad \text{height}=10-2=8\\ &\Rightarrow\ \text{hypotenuse}=\sqrt{6^2+8^2}=10 \end{aligned}
2

Apply the right-triangle inradius formula.

r=6+8−102=42=2\begin{aligned} &r=\frac{6+8-10}{2}=\frac{4}{2}=2 \end{aligned}
r=2r=2

General check: area =12⋅6⋅8=24=\tfrac12\cdot6\cdot8=24, semiperimeter s=6+8+102=12s=\tfrac{6+8+10}{2}=12, so r=Areas=2412=2r=\tfrac{\text{Area}}{s}=\tfrac{24}{12}=2.

CAT 2024 Slot 2 QA Q10: The coordinates of the three vertices of a triangle are: (1, 2), (7, 2), and (1, 10). Then the radius of the i — Solution | TheCATExam