# Math

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Trigonometry problem:

The sides AB and AC of triangle ABC are of lengths 4 and 7 respectively. M is the midpoint of BC. AM is of length 3.5, Find the length of BC.

• Math -

Complete the parallelogram ABPC , where BP=7 and PC=4
In a ||gram , diagonals bisect each other, so M is the midpoint of both diagonals and AP = 2(3.5) = 7

(looks like we have isosceles triangles, but that doesn't matter)

let angle PAC =Ø
let MC = a , thus BC = 2a for later

In triangle APC, by cosine law
4^2 = 7^2 + 7^2 - 2(7)(7)cosØ
cosØ = 82/98 = 41/49

in triangle AMC
a^2 = 3.5^2 + 7^2 - 2(3.5)(7)cosØ
= 61.25 - 49(41/49) = 20.25
a=√20.25 = 4.5

So BC = 2a = 2(4.5) = 9

better check my arithmetic, I have made some silly errors lately.

• Math -

Thank You!

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