CAT Practice : Mensuration

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Find the Ratio of two Triangles inside a sqaure.

Square and Triangle

Q.9: A square PQRS has an equilateral triangle PTO inscribed as shown:

What is the ratio of A∆PQT to A∆TRU?

1. 1 : 3
2. 1 : √3
3. 1 : √2
4. 1 : 2

Choice D. 1 : 2

Detailed Solution

Let PQ, a side of equilateral triangle be b

By symmetry QT=ST=z (say)

A ∆ PQT = ${\frac{1}{2} }$ x PQ x QT

= ${\frac{1}{2} }$ x a x z

A ∆ TRU = ${\frac{1}{2} }$ x RT x OR

= ${\frac{1}{2}}$ x (a-z) x (a-z)

${\frac{A ∆ PQT}{A ∆ TRU }}$ =${\frac{\frac{1}{2}*az}{\frac{1}{2}*(a-z)^2 }}$

= ${\frac{az}{(a-z)^2 }}$ --------------(p)

Since PQT and PTO are right angled triangles

PQ2 + QT2 = PT2

RT2 +RU2 = UT2

a2 + z2 = b2 --------- (1)

And, (a-z)2 + (a-z)2 = b2--------- (2)

=) a2 + z2 = 2(a-z)2

=) a2 + z2 = 2a2 + 2z2 – 4az

=) a2 + z2- 4az = 0

=) a2 + z2 – 2az = 2az (Please note how the solution is being managed here. You must always be aware of what you are looking for. Here, as equation -℗ we are looking for (a-z)2 in terms of az)

=) (a-z)2 =2az

Putting in equation (p) = ${\frac{az}{2(az) } = \frac{1}{2}}$

Choice (D) is therefore, the correct answer.

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Mensuration is the science of measurement. Cylinders, cones, cuboids, rectangular parallelopipeds etc.