Question

The two isosceles right-angle triangles shown in the figure are similar triangles. If the ratio of the area of the triangles is 1:3, what is x in cm?

Answer 1

Answer:

let the third angle be x

1x+x+3x equal to 90

5x equal to 90

x equal to 90/5

x equal to 18

Answer 2

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Let ABC,PQR be two isosceles triangle with equal vertical angles.

Let in ΔABC,AB=AC

⇒∠B=∠C=2180−∠A

And in ΔPQR,PQ=PR

⇒∠Q=∠R=2180−∠P

Given two vertical angles are equal.

∴∠A=∠P

⇒∠B=∠C=∠Q=∠R

By AAA postulate, "two triangles are similar if they have three corresponding angles congruent." 

∴ΔABC∼ΔPQR

We know ratio between the areas of two similar triangles is same as the ration between the squares of their corresponding altitudes and corresponding heights of two given triangles are AD and PS.

⇒Area of ΔABC/ΔPQR = AD²/PS²

⇒9/16 = AD²/PS²

⇒ AD : PS = 3 : 4

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