Question

triangle wxy is formed by connecting the midpoints of the side of triangle tuv. the lengths of the sides of triangle wxy are shown. find the perimeter of triangle tuv. figures not necessarily drawn to scale.

Explanation:

Step1: Recall Midsegment Theorem

The Midsegment Theorem states that the segment connecting the midpoints of two sides of a triangle is parallel to the third side and half as long. So, if \( W, X, Y \) are midpoints, then \( WX=\frac{1}{2}TV \), \( WY = \frac{1}{2}UV \), \( XY=\frac{1}{2}TU \).

Step2: Find lengths of \( TU, UV, TV \)

From the triangle \( WXY \), \( WY = 5 \), \( XY = 4 \), \( WX = 4 \).

• For \( TU \): Since \( XY=\frac{1}{2}TU \), then \( TU = 2\times XY=2\times4 = 8 \).
• For \( UV \): Since \( WY=\frac{1}{2}UV \), then \( UV = 2\times WY=2\times5 = 10 \).
• For \( TV \): Since \( WX=\frac{1}{2}TV \), then \( TV = 2\times WX=2\times4 = 8 \).

Step3: Calculate perimeter of \( \triangle TUV \)

Perimeter \( P=TU + UV+TV \). Substitute the values: \( P = 8 + 10+8=26 \).

Answer:

26