Right angles in triangles formed in trapezoid

Question

In the following trapezoid, is angle A in triangle ABC a right angle even though it isn't labeled as such? And if so what property would we use to determine that? I was able to get the correct area for triangle ABC using Heron's formula after finding side AC using the Pythagorean theorem. Therefore, angle A in triangle ABC must be a right angle, but I can't find the property.

Answer 1

Yes, it is. Because by Pythagorean Theorem on $\Delta AEB$, $|EB| = 9$ and therefore $|CE| = 25 - 9 = 16$. Then by Pythagorean Theorem on $\Delta AEC$, $|AC| = 20$. Then $\Delta ABC$ has sides $15, 20, 25$ which is a Pythagorean triple. Therefore, $\angle CAB = 90^\circ$.

Answer 2

From the information given it is easy to assess the claim.

First render $|EB|=9$ with the Pythagorean Theorem on $\triangle ABE$. Then render $|CE|=16$ by difference along $\overline{CB}$. Apply the Pythagorean Theorem to $\triangle ACE$ to get $|AE|$ and then check your result against the Pythagorean Theorem in $\triangle BCA$. You should find that the claim is true.

Bonus: Draw the perpendicular to $\overline{CB}$ from $D$ which intersects $\overline{CB}$ at $F$ prove that $|DA|=11$ is also correct.

Answer 3

If $$AE^2 = CE\times EB$$

then the triangle $ABC$ is a right triangle and the angle $CAB =90$ degrees.

We have $AE^2 =144$ and it is easy to see that $CE\times EB =16 \times 9 =144$$

Thus the answer is yes.