Perimeters of two similar triangles ABC and PQR  are in the ratio 4 : 5. If the sum of their areas is 164 cm2, find the area of each triangle.

Asked by Topperlearning User | 3rd Oct, 2017, 01:30: PM

Expert Answer:

Since triangles ABC and PQR  are similar ,

begin mathsize 12px style So comma space AB over PQ equals BC over QR equals AC over PR equals straight k space left parenthesis say right parenthesis
rightwards double arrow fraction numerator AB plus BC plus AC over denominator PQ plus QR plus PR end fraction equals straight k end style

So according to question straight k equals 4 over 5.

Let a side of triangle ABC be 4y and that of PQR is 5y.

We know that for two similar triangles the ratio of areas is equal to the ratio of squares of their corresponding sides.

Let the area of triangle ABC  is S and that of triangle PQR is S’.

begin mathsize 12px style So comma space fraction numerator straight S over denominator straight S apostrophe end fraction equals fraction numerator 16 straight y squared over denominator 25 straight y squared end fraction. space But comma space straight S plus straight S apostrophe equals 164 space cm squared space space space left parenthesis given right parenthesis
So comma space 16 straight y squared plus 25 straight y squared equals 164 space cm squared
rightwards double arrow 41 straight y squared equals 164 space cm squared
rightwards double arrow straight y squared equals 4 space rightwards double arrow straight y equals 2 space space
So comma space the space area space of space increment ABC space is space 64 space cm squared space and space the space area space of space increment PQR space is space 100 space cm squared. end style

Answered by  | 3rd Oct, 2017, 03:30: PM