Question

consider $\triangle abc$. what is the length of $\overline{ac}$? \
a. 32 units \
b. 24 units \
c. 48 units \
d. 16 units

Explanation:

Step1: Identify Midline Theorem

In triangle \( \triangle ABC \), \( G \) is the midpoint of \( AB \) (since \( AG = GB \)) and \( H \) is the midpoint of \( BC \) (since \( BH = HC \)). By the Midline Theorem (Midsegment Theorem), the segment connecting the midpoints of two sides of a triangle is parallel to the third side and half its length. So \( GH \) is the midline, and \( GH=\frac{1}{2}AC \).

Step2: Solve for \( AC \)

Given \( GH = 16 \) units. From the Midline Theorem, \( GH=\frac{1}{2}AC \), so we can solve for \( AC \) by multiplying both sides by 2: \( AC = 2\times GH \). Substituting \( GH = 16 \), we get \( AC = 2\times16 = 32 \) units.

Answer:

A. 32 units