CAT Quantitative Aptitude Questions | Geometry Questions for CAT - Triangles

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Explanatory Answer

Method of solving this CAT Question from Triangles: Apollonius’s theorem comes in handy in situations.

E is the midpoint of hypotenuse AC so, BE = 1/2 ∗ AC=50 cm
Now, applying Apollonius’s theorem on ΔABE
=> AB² + BE² = 2 * (BD² + AD²)
=> AB² + 50² = 2 * (BD² + 25²)……….(1) (As, AD = DE = EF = FC = 25 cm)

Similarly, in ΔBEC,
=> BC² + BE² = 2 * (BF² + EF²)
=> BC² + 50² = 2 * (BF² + 25²)………..(2)

Adding (1) and (2),
=> AB² + BC² + 2 * 50² = 2 * (BD² + BF² + 2 * 25²)
=> AC² + 2 * 50² = 2 * (BD² + BF²) + 4 * 25² (As, AB² + BC² = AC²)
=> 100² + 2 * 50² – 50² = 2 * (BD² + BF²)
=> 100²+ 50² = 2 * (BD² + BF²)
=> (10,000 + 2500)/2 = (BD² + BF²)
=> 12500/2 = BD² + BF²
Therefore, BD² + BF² = 6250
=> BD² + BF² + BE² = 6250 + 50² = 6250 + 2500 = 8750 cm²

The question is "Find the value of: BD² + BE² + BF² (in cm²)"

Hence, the answer is 8,750 cm².

Choice C is the correct answer.