Finding $AC$ given that $ABCD$ is inscribed in a circle

Question

Suppose quadrilateral $ABCD$ is inscribed in a circle such that $AB=4, BC=5, CD=6,$ and $DA=7.$ Find the length of $AC.$

I do realize that this is a repost, but the previous asker and answerers did not state a clear method as to how to do this problem. I believe Ptolemy's theorem would be useful, but I am unsure how to apply it. Can someone give me a hint please?

JCAA · Answer

Use the cosine theorem for triangles ABC and ADC. The angles ABC and ADC add up to 180. Denote the angle ABC by $X$. Then
$$4^2+5^2 - 2*4*5cos X=
6^2+7^2+2*6*7*cos X,$$
so $124cos X = -44$. So $cos X= -44/124=-11/31$, then $AC^2=4^2+5^2-40*11/31$.

Finding $AC$ given that $ABCD$ is inscribed in a circle

One Answer

Add your own answers!

Ask a Question