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XAT 2012 — QA & DI Question 13

Geometric CentersEasy
Passage / Data

Answer the following question based on the information given below.

The following graphs shows the revenue (in $ million) of three companies in their initial six years of operations, in an economy which is characterized by a persistent inflation.

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A city has a park shaped as a right angled triangle. The length of the longest side of this park is 80 m. The Mayor of the city wants to construct three paths from the corner point opposite to the longest side such that these three paths divide the longest side into four equal segments. Determine the sum of the squares of the lengths of the three paths.

Answer & solution

  • A

    4000 m

  • B

    4800 m

  • 5600 m

  • D

    6400 m

  • E

    7200 m

Solution

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​​​Here D is midpoint of AC, E is the midpoint of AD and F is the midpoint of CD.

Hence, AE = ED = DF = CF = 20

Let AB = c and BC = a

Applying Apollonius theorem in ∆ABC, we get,

BD2 + 402 = 1/2 × (c2 + a2) = 1/2 × 802

BD2 = 402 … (i)

Now, applying Apollonius theorem in ∆ABD, we get,

BE2 + 202 = 1/2 × (c2 + BD2) … (ii)

Similarly, applying Apollonius in ∆CDB, we get,

BF2 + 202 = 1/2 × (a2 + BD2) … (iii)

Adding (ii) and (iii), and substituting value from (i), we get,

BE2 + BF2 + 2 × 202 = 1/2 (a2 + c2 + 2 × BD2) = 1/2 (802 + 2 × 402) = 3 × 402

Hence, BE2 + BF2 + BD2 = 3 × 402 + 402 – 2 × 202 = 5600

Hence, option (c).

XAT 2012 QA & DI Q13: A city has a park shaped as a right angled triangle. The length of the longest side of this park is 80 m. The — Solution | TheCATExam