What is the length of segment AC?

Answer: 5 units

The two right triangles are similar because they share a pair of vertical angles (∠ACB and ∠ECD). Vertical angles are always congruent. Obviously both right angles (∠B and ∠D) are congruent. Thus, angles A and E are congruent because of the triangular sum theorem.

With similar triangles, corresponding sides will be proportional. Segment \(\overline{\mathit{BC}}\) is half the length of \(\overline{\mathit{CD}}\), therefore \(\overline{\mathit{AC}}\) will be half the length of \(\overline{\mathit{CE}}\). The length of \(\overline{\mathit{CE}}\) can be computed from the Pythagorean theorem, since it is the hypotenuse of a right triangle for which the lengths of the other two sides are known:

\(\overline{\mathit{CE}}=\sqrt{6^{2}+8^{2}}=\sqrt{100}=10\)

The length of \(\overline{\mathit{AC}}\) will be half of this value, or 5 units.