Given: RL = 14, RK = 12, KL = 10. In the figure, M is the midpoint of RL, N is the midpoint of KL, and T is the midpoint of RK. Find x and y.

Solution:

We are given a triangle RKL with sides RL = 14, RK = 12, and KL = 10. Points M, N, and T are midpoints of RL, KL, and RK respectively. We need to find the lengths of segments x and y.

Finding the length of segment x:

Segment x is the segment MT. By the Midpoint Theorem, the segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half the length of the third side.

1. M is the midpoint of RL, and T is the midpoint of RK. Therefore, MT is parallel to KL and MT = \(\frac{1}{2}\) KL.
2. Given KL = 10, so MT = \(\frac{1}{2}\) \( \times 10 = 5 \).
3. Thus, x = 5.

Finding the length of segment y:

Segment y is the segment MN. By the Midpoint Theorem:

1. M is the midpoint of RL, and N is the midpoint of KL. Therefore, MN is parallel to RK and MN = \(\frac{1}{2}\) RK.
2. Given RK = 12, so MN = \(\frac{1}{2}\) \( \times 12 = 6 \).
3. Thus, y = 6.

The segment connecting T and N is TN. T is the midpoint of RK and N is the midpoint of KL. Therefore, TN is parallel to RL and TN = \(\frac{1}{2}\) RL.

Given RL = 14, so TN = \(\frac{1}{2}\) \( \times 14 = 7 \).

The figure shows that x represents the length of segment MT and y represents the length of segment MN.

Answer: x = 5, y = 6.