Pythag Bash?

Geometry Level 2

Suppose triangle $ABC$ has side lengths $AB = 4, AC = 5,$ and $BC = 6$ . Let the perpendicular from $A$ go past line $BC$ to point $D$ such that $\angle BDC$ is $90^{\circ}$ . Find $16\cdot [BDC]^{2}$ where $[BDC]$ denotes the area of triangle $BDC$ .

The answer is 1215.

1 solution

Daniel Stewart
Sep 2, 2017

Note $ABEC$ is orthodiagonal so if $x = BE$ and $y = EC$ , we have $y^{2}-x^{2} = 9$ and by the Pythagorean Theorem $x^{2} + y^{2} = 36.$ This is a linear system in $x^2$ and $y^2$ . Solving for those, we get $x^2 = \frac{27}{2}$ and $y^{2} = \frac{45}{2}$ . Since the question is asking for $2x^{2}y^{2},$ we compute the answer to be $\boxed{1215}.$ Mention of there are any errors.

Pythag Bash?

The answer is 1215.

1 solution

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