I've been practicing problems recently to study for the upcoming AMC10, and came across one I could not figure out how to solve one of them. The diagram of it is attached below, and the problem goes a little like this.
Quadrilateral PTSU is inscribed in semicircle O, as shown, with PQ = 3 units, QR = 5 units, and RS = 4 units. What is the area of PSTU?
I haven't been able to find a way to get to the area. I'm not completely sure how to find the length of the other sides, and I'm not aware of any method I can use for this quadrilateral to easily find the area of it. Any ideas?
Best Answer
The diameter is $3+5+4=12$ and the radius is thus $12/2=6$.
You can connect $OT$ and $OU$ as auxiliary lines. Then we have $|OR|=|OS|-|RS|=6-4=2$ and $$|RT|=\sqrt{|OT|^2-|OR|^2}=\sqrt{6^2-2^2}=4\sqrt{2}.$$ Similarly, we have $|OQ|=|OP|-|PQ|=6-3=3$ and $$|UQ|=\sqrt{|OU|^2-|OQ|^2}=\sqrt{6^2-3^2}=3\sqrt{3}.$$ Then you can calculate the area immediately.