Question

Find the lengths of the sides of the cylcic quadrilateral if one diagonal coincides with a diameter of a circle whose area is 36pi cm squared. The other diagonal measures 8 cm meets the fist diagonal at right angles.

Answer 1

Since BD is a diameter, angle A = angle C = 90 degrees.

In cyclic quadrilateral, opposite angles add to 180 degrees:
angle B + angle D = 180 degrees.

Answer 2

Ah, yes. That's one of the theorems.

Answer 3

q^2 + r^2 = 12^2 --- (1)
p^2 + s^2 = 12^2 --- (2)
Four unknowns and so far we have two equations. We need two more.

Answer 4

oh, ok. I get it.

Answer 5

Oh, I just noticed the line "The other diagonal measures 8 cm meets the fist diagonal at RIGHT ANGLES." We may not need the last diagram. Let us try this:

Answer 6

r^2 + q^2 = 12^2 = 144 ---- (1)
t^2 + u^2 = q^2 ---- (2)
t^2 + (12-u)^2 = r^2 ---- (3)
(12-u) * u = t * (8-t) ---- (4)

Four equations and four unknowns. Eliminate t and u and solve for q and r.

Then repeat the same procedure for the bottom half.

Answer 7

Ah, ok I'll start working with it.

Answer 8

http://www.wolframalpha.com/input/?i=r^2+%2B+q^2+%3D+12^2+%3D+144%2C++t^2+%2B+u^2+%3D+q^2%2C+t^2+%2B+%2812-u%29^2+%3D+r^2+%2C+%2812-u%29+*+u+%3D+t+*+%288-t%29%2C+q%3E0%2Cr%3E0%2Ct%3E0%2Cu%3E0+

Answer 9

Even though it looks like there are two sets of solutions from the above link, there is only one set with just q and r values interchanged. So pick just one set of solutions for q and r.

Since we know t and u, we can find s and p as follows:
s^2 = (12-u)^2 + (8-t)^2. Put values for u and t and calculate s.
p^2 = u^2 + (8-t)^2. Put values for u and t and calculate p.

Answer 10

err, I'm trying to get the values xD

Answer 11

Welp, I couldn't figure it out. I'll just wait for our prof. to discuss it I guess. Thank you very much aum.

Answer 12

A simpler set of equations seems to result from similar triangles.