An isosceles triangle has sides A, B, and C, such that sides A and B have the same length. Side C has a length of $56$ $56$ and the triangle has an area of $112$ $112$ . What are the lengths of sides A and B?

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Answer 1 · 2018-05-12T00:08:30

$20 \sqrt{2}$ $20 sqrt{2}$

Let's renotate and say triangle ABC has $a = b$ $a=b$ , $c = 56$ $c=56$ area $A = 112$ $A=112$ .

Archimedes' Theorem is a modern version of Heron's Formula,

$16 A^{2} = 4 a^{2} c^{2} - (b^{2} - a^{2} - c^{2})$ $16A^2 = 4 a^2 c^2 - (b^2 - a^2 - c^2)$

$16 {(112)}^{2} = 4 a^{2} {(56)}^{2} - 56^{4}$ $16(112)^2 = 4a^2 (56)^2 - 56^4$

$a = 20 \sqrt{2}$ $a = 20 sqrt{2}$

An isosceles triangle has sides A, B, and C, such that sides A and B have the same length. Side C has a length of 56 5656 and the triangle has an area of 112 112112 . What are the lengths of sides A and B?