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.
In the figure above, segment BC is 4 units long, segment CD is 8 units long, and segment DE is 6 units long. What is the length of segment AC?
- 4 units
- 5 units
- 6 units
- 8 units
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