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CAT 2024 Slot 1 QA Question & Solution

GeometryMedium

Question

ABCD is a rectangle with sides AB = 56 cm and BC = 45 cm, and E is the midpoint of side CD. Then, the length, in cm, of radius of incircle of $\triangle ADE$ is

Solution

Drawing the figure described in the question, we get the following representation, 

Screenshot 2024-12-04 at 10

Since E is the midpoint of CD, the length of each half will be 28cm. 

We need to find the incircle of the triangle ADE, which is a right angled triangle, we can do that with the formula, 
$\dfrac{\left(a+b-h\right)}{2}$

$h^2=28^2+45^2$
$h^2=784+2025=2809$
$h=53$

$r=\dfrac{\left(28+45-53\right)}{2}=10$