Answer to Question #334421 in Geometry for Angel

Question

A rectangle ABCD which measures 18 by 24 units is folded once, perpendicular to

diagonal AC, so that the opposite vertices A and C coincide. Find the length of the fold.

Answer 1

ABCD - rectangle

AB = CD = 18

BC = AD = 24

MN perpendicular of diagonal AC

AO = OC = AC/2

MO = ON = x

From the Pythagorean theorem AC² = AB² + BC²

$AC = \sqrt{AB^2 + BC^2}\\ AC = \sqrt{18^2 + 24^2} = 30,\:then\:OC = 30/2 = 15$

From the triagle ABC

$\angle BCA = \theta\\ tan\theta = \frac{AB}{BC} = \frac{18}{24} = \frac{3}{4}$

From the triagle MCO

$\angle MCO = \theta\\ tan\theta = \frac{OM}{OC} = \frac{x}{15}, then\\ \frac{x}{15} = \frac{3}{4}\\ 4x = 15\times3\\ x = 45/4 = 11.25,\: then\\ MN = 2\times x = 2\times11.25 = 22.5$

Answer:

Lenght of fold = 22.5