Question

art the divider bars between the pieces of colored glass in a stained glass window are called cames. in the stained window shown, the total length of the cames for $\triangle pqr$ is 78 centimeters. what is the total length of the cames for $\triangle jkl$? give an argument to support your answer. \boxed{} cm. because j, k, and l are midpoints of their respective sides, $jk = \frac{1}{2}\boxed{}$, $kl = \frac{1}{2}\boxed{}$, and $jl = \frac{1}{2}\boxed{}$ by the triangle midsegment theorem. so $jk + kl + jl = \frac{1}{2}(\boxed{}) = \boxed{}$

Explanation:

Step1: Identify Midsegment Relationships

J, K, L are midpoints, so by Triangle Midsegment Theorem, \( JK = \frac{1}{2}PR \), \( KL = \frac{1}{2}PQ \), \( JL = \frac{1}{2}QR \).

Step2: Sum of Sides for \( \triangle JKL \)

Perimeter of \( \triangle PQR \) is \( PQ + QR + PR = 78 \) cm.
Perimeter of \( \triangle JKL \) is \( JK + KL + JL = \frac{1}{2}(PR + PQ + QR) \).

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

Substitute \( PQ + QR + PR = 78 \):
\( \frac{1}{2} \times 78 = 39 \) cm.

Answer:

39 cm. Because \( JK = \frac{1}{2}PR \), \( KL = \frac{1}{2}PQ \), \( JL = \frac{1}{2}QR \) (by Triangle Midsegment Theorem), so \( JK + KL + JL = \frac{1}{2}(PQ + QR + PR) = \frac{1}{2} \times 78 = 39 \).