Question

in \\( \triangle x y z, \overline{a b} \\) is a midsegment. find the length of \\( \overline{y z} \\). (not drawn to scale) a 3 b 20 c 4 d 18

Explanation:

Step1: Use the midsegment property

Since \( \overline{AB} \) is a midsegment, \( XA = AY \). So \( 4x - 1=2x + 5 \).
Solve for \( x \):
\( 4x-2x=5 + 1 \)
\( 2x=6 \)
\( x = 3 \).

Step2: Find the length of \( \overline{AB} \)

Substitute \( x = 3 \) into \( AB=3x + 1 \).
\( AB=3\times3+1=9 + 1=10 \).

Step3: Use the midsegment - triangle side relationship

The midsegment theorem states that the length of the midsegment \( AB \) is half the length of \( YZ \). Let \( YZ = l \), then \( AB=\frac{1}{2}YZ \).
Since \( AB = 10 \), then \( l=2\times AB \).
\( YZ=2\times10 = 20 \).

Answer:

B. 20