Question

if c is the midpoint of $overline{ab}$ and $overline{de}$, what is the length of $overline{ab}$? (1 point) 13 26 17 24

Explanation:

Step1: Find length of DC

Since C is the mid - point of DE and DE = 12, then DC=CE = 6.

Step2: Use Pythagorean theorem in triangle ADC

In right - triangle ADC, AD = 5 and DC = 6. By the Pythagorean theorem \(AC=\sqrt{AD^{2}+DC^{2}}=\sqrt{5^{2}+6^{2}}=\sqrt{25 + 36}=\sqrt{61}\).

Step3: Find length of AB

Since C is the mid - point of AB, then AB = 2AC. In right - triangle BCE, BC has the same length as AC (by congruence of right - triangles ADC and BCE, as \(\angle ADC=\angle BEC = 90^{\circ}\), DC = CE and \(\angle ACD=\angle BCE\)). So \(AB = 2\sqrt{5^{2}+12^{2}}\).
\[

$$\begin{align*} AB&=2\sqrt{25 + 144}\\ &=2\sqrt{169}\\ &=2\times13\\ &=26 \end{align*}$$

\]

Answer:

26