Question

If the area of two similar triangles are in the ratio 16:25,then find the ratio if thier corresponding sides

Answer 1

Let’s say the first triangle ABC has sides of length a,b,c, and the second triangle DEF is similar, i.e. ABC ~ DEF. The definition of similarity says its sides are proportional to those of ABC and the triangles have equal angles, so there are two sides with lengths ka and kb in DEF for whom the angle in between (let’s call it F) equals that between a and b in ABC, which we’ll call C.

Using the triangle area formula 1/2absinC, we can write the ratio of areas ABC and DEF as 1/2absinC1/2(ka)(kb)sinF=1/k2=16/251/2absinC1/2(ka)(kb)sinF=1/k2=16/25 by canceling out equal angles and variables. This tells us that k = 5/4, or that ABC and DEF have triangle sides in ratio 4:5.


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