Rhombus ADEF is inscribed into a triangle ABC so that they share angle A and the vertex E lies on the side BC . What is the length of the side of the rhombus if AB=c, and AC=b.

Question

Rhombus ADEF is inscribed into a triangle ABC so that they share angle A and the vertex E lies on the side 

BC

. What is the length of the side of the rhombus if AB=c, and AC=b.



Answer 1

triangle BDE is similar to triangle BAC

corresponding sides of similar triangles form equal ratios

BD/BA=DE/AC

AD=DE=EF=FA because the sides of a rhombus are all equal in measure

AB=c and AC=b

BD/BA becomes...

BD=c-AD and BA=c

so, (c-AD)/c is one ratio

DE/AC becomes...

DE=AD because of the above (you have a rhombus with equal sides), and AC=b

so, AD/b is another ratio

set them equal to each other

(c-AD)/c=AD/b

cross multiply

b(c-AD)=cAD

bc-ADb=ADc

bc=ADb+ADc

bc=AD(b+c)

AD=bc/(b+c) is the length of each side of the rhombus

Answer 2

Hi Evan J.,

Yeah, no problem. You can see that s must be related to b and c. So how?

Since the problem gives you similar triangles, use them!

Start writing proportions in the similar triangles:

FC/AC = FC/b = s/c (similar triangle corresponding sides)

also FC = b - s (since they are adjacent collinear segments)

so FC = bs/c

Then equate the two representations: FC = FC; b - s = bs/c ; b = s (1 +b/c) ; s = b / (1 +b/c)

solves to s = bc/(b+c)

Reality check: s must be smaller than both b and c, since b+c (denominator) is larger than either b or c (numerator factors), which makes sense in the figure!

-- Cheers, -- Mr. d.