Triangle Bonanza.

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In right A B C \triangle{ABC} , A D \overline {\rm AD} is the altitude to hypotenuse B C \overline {\rm BC} with A D = c |\overline {\rm AD}|= c and B C = 5 2 c |\overline {\rm BC}| = \dfrac{5}{2}c .

If in right A B C \triangle{ABC} the perimeter P = c 2 P = c^2 and the value of c c can be expressed as

c = d e + e f c = \dfrac{d * \sqrt{e} + e}{f} , where d , e d,e and f f are coprime positive integers, find d + e + f d + e + f .


The answer is 10.

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1 solution

Rocco Dalto
Dec 2, 2019

For convenience let j = 5 2 j = \dfrac{5}{2} (This is how I constructed the problem).

Note: j > = 2 j >= 2 ensures that A B C \triangle{ABC} is a right triangle.

In A B C \triangle{ABC} the altitude is the mean proportion between the segments of the hypotenuse

c 2 = x ( j c x ) = ( j c ) x x 2 x 2 ( j c ) x + c 2 = 0 \implies c^2 = x(jc - x) = (jc)x - x^2 \implies x^2 - (jc)x + c^2 = 0 \implies

x = ( j ± j 2 4 2 ) c x = (\dfrac{j \pm \sqrt{j^2 - 4}}{2})c

Using x = ( j ± j 2 4 2 ) c D C = ( j j 2 4 2 ) c x = (\dfrac{j \pm \sqrt{j^2 - 4}}{2})c \implies |\overline {\rm DC}| = (\dfrac{j - \sqrt{j^2 - 4}}{2})c and B D = ( j + j 2 4 2 ) c |\overline {\rm BD}| = (\dfrac{j + \sqrt{j^2 - 4}}{2})c

A C = A D 2 + D C 2 = j 2 j j 2 4 2 c \implies |\overline {\rm AC}| = \sqrt{ |\overline {\rm AD}|^2 + |\overline {\rm DC}|^2} = \sqrt{\dfrac{j^2 - j\sqrt{j^2 - 4}}{2}} c

and

A B = B D 2 + A D 2 = j 2 + j j 2 4 2 c |\overline {\rm AB}| = \sqrt{ |\overline {\rm BD}|^2 + |\overline {\rm AD}|^2} = \sqrt{\dfrac{j^2 + j\sqrt{j^2 - 4}}{2}}c

Note: Using x = ( j j 2 4 2 ) c x = (\dfrac{j - \sqrt{j^2 - 4}}{2})c the values of A B |\overline {\rm AB}| and A C |\overline {\rm AC}| are just switched, so that A C = j 2 + j j 2 4 2 |\overline {\rm AC}| = \sqrt{\dfrac{j^2 + j\sqrt{j^2 - 4}}{2}} and A B = j 2 j j 2 4 2 c |\overline {\rm AB}| = \sqrt{\dfrac{j^2 - j\sqrt{j^2 - 4}}{2}} c .

Using j = 5 2 j = \dfrac{5}{2} The Perimeter P = ( 3 5 + 5 2 ) c = c 2 P = (\dfrac{3\sqrt{5} + 5}{2})c = c^2 \implies

c ( c ( 3 5 + 5 2 ) ) = 0 c(c - (\dfrac{3\sqrt{5} + 5}{2})) = 0 and c 0 c = 3 5 + 5 2 = c \neq 0 \implies c = \dfrac{3\sqrt{5} + 5}{2} =

d e + e f d + e + f = 10 \dfrac{d * \sqrt{e} + e}{f} \implies d + e + f = \boxed{10} .

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