Question

triangle rst is formed by connecting the midpoints of the side of triangle opq. the lengths of the sides of triangle opq are shown. find the perimeter of triangle rst. figures not necessarily drawn to scale. image of triangle opq with midpoints r, s, t forming triangle rst; side oq is 12, side pq is 10, side op is 8

Explanation:

Step1: Recall Midline Theorem

The midline (or midsegment) of a triangle is parallel to the third side and half as long. Since \( R, S, T \) are midpoints, \( RT \parallel PQ \), \( RS \parallel OQ \), \( ST \parallel OP \), and each midline is half the length of the corresponding side of \( \triangle OPQ \).

Step2: Find Side Lengths of \( \triangle RST \)

• Length of \( RT \): Corresponding to \( PQ = 10 \), so \( RT=\frac{10}{2} = 5 \).
• Length of \( RS \): Corresponding to \( OQ = 12 \), so \( RS=\frac{12}{2}=6 \).
• Length of \( ST \): Corresponding to \( OP = 8 \), so \( ST=\frac{8}{2} = 4 \).

Step3: Calculate Perimeter of \( \triangle RST \)

Perimeter \( = RT + RS + ST = 5 + 6 + 4 = 15 \).

Answer:

15