Question

points q and r are midpoints of the sides of triangle abc. what is aq?
○ 10 units
○ 14 units
○ 20 units
○ 32 units
(diagram: triangle abc with q on ab, r on ac; aq labeled 4p, ar labeled 2p + 3, bc labeled 6p - 4; tick marks indicate aq = qb and ar = rc)

Explanation:

Step1: Apply Midline Theorem

Since Q and R are midpoints, \( QR \) is a midline, so \( QR=\frac{1}{2}BC \) and \( AQ = QB \), also \( AR = RC \). From the diagram, \( QR = 2p + 3 \) and \( BC=6p - 4 \). By midline theorem, \( 2(2p + 3)=6p - 4 \).

Step2: Solve for \( p \)

Expand left side: \( 4p + 6 = 6p - 4 \).
Subtract \( 4p \) from both sides: \( 6 = 2p - 4 \).
Add 4 to both sides: \( 10 = 2p \).
Divide by 2: \( p = 5 \).

Step3: Find \( AQ \)

\( AQ = 4p \), substitute \( p = 5 \): \( AQ = 4\times5 = 20 \).

Answer:

20 units