A piece of paper has the precise shape of an equilateral triangle ABC which is creased along a line XY, with X on AB and Y on AC. so that A falls on some point D on BC.

(i) Are the triangles XBD and DCY always similar?
If so, prove it. If not, give a counterexample.

(ii) If AB = 15 and BD = 3, what is the length of the crease XY?

Tomarken established part (i)

Note triangles AXY and DXY are congruent, as they are reflections.
Call CY=x by similar triangles BX=36/x
AY=DY=15-x
AX=DX=15-36/x
So using the similar triangles again
x/3 = (15-x)/(15-36/x)
Solving gives x=4.5
so
AY = 10.5
AX = 7
And using the law of cosines XY˛=85.75
XY = √85.75 ≈ 9.260129589

Posted by Jer on 2014-03-06 15:34:49