CAT 2024 Slot 1 · Quant · Question 19
CAT 2024 Slot 1 previous year question on Triangles, with the answer and a step-by-step solution.
Answer & solution
Answer: 10
In the rectangle, CD=AB=56 and AD=BC=45.
E is the midpoint of CD, so DE=\frac{56}{2}=28.
The angle at D is a corner of the rectangle, so triangle ADE is right-angled at D with legs AD=45 and DE=28.
Its hypotenuse is AE=\sqrt{{45}^{2}+{28}^{2}}=\sqrt{2025+784}=\sqrt{2809}.
Since {53}^{2}=2809, we get AE=53.
For a right triangle the inradius is r=\frac{\text{leg}+\text{leg}-\text{hypotenuse}}{2}.
Substituting, r=\frac{45+28-53}{2}=\frac{20}{2}=10 cm.
Shortcut. The right-triangle formula r=\frac{a+b-c}{2} avoids computing the area and the semi-perimeter separately. Spotting that \left( 28,45,53 \right) is a Pythagorean triple removes the square root too.
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