Find the height of the trapezoid - is there enough information?

Question

I have an isosceles trapezoid inscribed in a semicircle. Let the longer base of the trapezoid be the diameter of the circle, call it AB, with measure of 20 inches. The congruent legs are the chords AC and BD, each with measure of 12 inches. The goal of the problem is to find the height of the trapezoid. However, I'm skeptical about whether there is even enough information here. I thought I could possibly draw CB splitting the trapezoid into two triangles and use Heron's formula or law of cosines, but not enough information is given for that. And then I thought, something as simple as the pythagorean theorem, could work, but again, not enough information is given. I thought I could combine these ideas and write a system of equations, but I keep ending up with more variables than equations. Is this problem even solvable?

Answer 1

Let $D=(u,v)$ with $u>0$, $v>0$. Then $u^2+v^2=100$, $(10-u)^2+v^2=144$. From these two equations you an obtain a linear equation for $u$.

Answer 2

There is certainly enough information.

As it is inscribed in a circle there are are only two points of the circle that are a distance of $12$ from an endpoint of the diameter. Arbitrarily choosing the ones that are "above the diameter" make the two vertices of the top of the trapezoid unique.

To put these an a coordinate plane the four vertices are are $A= (-10,0); B=(10,0)$ and $C = (-u, v); D=(u,v)$. (Because it is an isosceles trapezoid $\overline {CD} $ is parallel to $\overline{AB}$ so the the $y$ coordinates of $C,D$ are equal and the $x$ coordinates are centered around the $y$ axis).

The circle has formula $x^2 + y^2 = 10$ so $u^2 + v^2 = 10^2$. And the distance $BD = \sqrt{(u-10)^2 + (v-0)^2} = 12$.

So solve for $u$ and $v$ and the height will be the value of $v$.