Solution to puzzle 23: Length of hypotenuse

Triangle ABC is right-angled at B. D is a point on AB such that BCD = DCA. E is a point on BC such that BAE = EAC. If AE = 9 inches and CD = 8 inches, find AC.

Following is one method of solution; others exist.

Let BAE = x, and BCD = y.

AB = 9 cos x = AC cos 2x.
BC = 8 cos y = AC cos 2y.

Eliminating AC:
(9 cos x) / (cos 2x) = (8 cos y) / (cos 2y).

y = 45° − x. Also, 2y = 90° − 2x, and so cos 2y = sin 2x. Therefore:
(9 cos x) / (cos 2x) = (8 cos (45°−x)) / (sin 2x).

Using trigonometric identity cos(a − b) = cos a · cos b + sin a · sin b:
cos (45°−x) = (cos x + sin x) /.

Rearranging: tan 2x = 8(cos x + sin x) / 9 cos x.

Using trigonometric identity tan 2a = 2 tan a / (1 − tan²a), and letting t = tan x:
2t / (1 − t²) = 8(1 + t)/9.
Therefore 9t/4 = (1 + t)(1 − t²) = 1 + t − t² − t³.
Hence t³ + t² + (5/4)t − 1 = 0.

By inspection, one root is t = 1/2.
Therefore (t − 1/2)(t² + 3t/2 + 2) = 0.
The quadratic factor has no real roots (since (3/2)² − 4·1·2 < 0), and so t = 1/2 is the only real root.

AC = 9 · cos x / cos 2x.

Using trigonometric identities cos x = 1 / (1 + t²), cos 2x = (1 − t²)/(1 + t²):
AC = 9 · (1 + t²) / (1 − t²).

Therefore AC = 9 · (/2) / (3/4) = 6 inches.

Source: Based on message 5934 in the Hyacinthos Yahoo! Group