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If the ratio of the perimeter of two similar triangles is 9:16, then what is the ratio of the area?
We can use the area theorem of the triangle to solve the above question.
Answer: The ratio of the area will be 81:256
Let's proceed step by step.
Let us consider ∆ABC whose sides are a, b and c respectively and ∆PQR whose sides are p, q and r respectively.
Let ∆ABC and ∆PQR be two similar triangles.
Assume k as a constant.
So, a/p = b/q = c/r = k
Hence , a = pk , b = qk and c = rk.
(a + b + c) / (p + q + r) = 9/16
Substituting values of a, b and c: a = pk , b = qk and c = rk, we get
k.(p + q + r) / (p + q + r) = 9/16. [a + b + c = k ( p+q+r)]
Area of ∆ABC / Area of ∆PQR = (a/p)2 = (b/q)2 = (c/r)2 = k2.
= 81 / 256
= 81 : 256
So, the ratio of the area is 81 : 256