I am reviewing trigonometry and this question has puzzled me.

isosceles triangle $\displaystyle PQR$ has a base of $\displaystyle QR$ 10cm long. The height is 4cm. Determine the measures if the three angles the given triangle.

The section I'm on is using TAN. So what I did is draw a perpendicular bisector, bisection the height $\displaystyle PQ$ so I now have right angles so I can use trigonometry "it has to be a right angle triangle right?" anywho, then I used pythagorus to find the length of the perpindicluar bisecter which came out to 9.79cm.

Then I found the angles of the two equal angles "angles P and Q" and I got 78°, so the remaining angle would be 24°.

The answer key has 38.7°, 38.7°, and 102.6°

What am I doing wrong, am I making it harder than it needs to be?

Thanks, any help is greatly appreciated.