Two isosceles Triangles have equal vertical angles and their

This question trumped me the first time i attempted it, because SIMILARITY and properties of SIMILAR triangles are easy to forget, but they are very important, as i have seen quite a few questions where knowledge of similarity is important, especially when we have one big triangle being divided into two etc etc (MGMAT questions, in their advanced book) ..

In addition to the explanation given by Bunuel, this link is worth bookmarking :

https://www.mathopenref.com/similartriangles.html

You can use the property of two similar triangles : If two similar triangles have corresponding side lengths in ratio a:b, then their areas will be in ratio a\(2\) : b\(2\)

I have similar solution:

Area1: 16=1/2*a*h => a=32/h
Area2: 25=1/2*A*H => A=50/H

Find: h/H

Similar triangles so: h/H=a/A

Solve:

h/H=a/A

h/H=(32/h)/(50/H)

h/H=(32/h)*(H/50)

h/H=(32/50)*(H/h)

h/H=(16/25)*(H/h)

h^2/H^2=16/25

h/H=4/5

Answ: A.

@Bunuel- Dear Sir- How do you know that the triangles are similar?

Given

• Two isosceles Triangles have equal vertical angles and their areas are in the ratio 16:25.

To Find

• The ratio of their corresponding heights.

Approach and Working Out

• The ratio of area = 16/25

• Area is 2-dimensions and the height one-dimension.

• They are similar so the ratio of any of the corresponding sides, heights, or even perimeter (all that are 1D) will be in the ratio \(\sqrt{16/25}\).

o 4/5

Correct Answer: Option A

The question uses language you will certainly never see in a real GMAT question. In an isosceles triangle (that is not equilateral), the "vertical angle" is the angle unequal to the other two (the angle between the two equal sides). The GMAT will never use this phrase, so you don't need to know it. But if two triangles are isosceles, and have the same "vertical angle", then the pair of equal angles in the two triangles will also be the same, because angles in a triangle sum to 180 (if the vertical angle is "x", the two equal angles will both be 90 - (x/2)). So the triangles have identical angles, and are thus similar triangles, and if their areas are in the ratio b to c, any corresponding lengths will necessarily be in the ratio √b to √c, by similarity, from which we get the answer 4 to 5.

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