Question

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

Explanation:

Step1: Use the midsegment property (A is the midpoint of XY, so \(AX = AY\))

Since \(AX = 4x - 1\) and \(AY = 2x + 5\), we set \(4x - 1 = 2x + 5\).

Step2: Solve for \(x\)

Subtract \(2x\) from both sides: \(4x - 2x - 1 = 2x - 2x + 5\) → \(2x - 1 = 5\).
Add 1 to both sides: \(2x - 1 + 1 = 5 + 1\) → \(2x = 6\).
Divide by 2: \(x = \frac{6}{2} = 3\).

Step3: Find the length of the midsegment \(AB\)

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

Step4: Use the midsegment theorem (\(AB=\frac{1}{2}YZ\))

So \(YZ = 2 \times AB = 2 \times 10 = 20\).

Answer:

B. 20