Question

triangle uvw is formed by connecting the midpoints of the side of triangle rst. the lengths of the sides of triangle rst are shown. what is the length of \\(\overline{uv}\\)? figures not necessarily drawn to scale.

Explanation:

Step1: Recall Midline Theorem

The Midline Theorem (also known as 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.

Step2: Identify the third side

In triangle \( RST \), \( U \), \( V \), \( W \) are midpoints. So, \( \overline{UV} \) connects midpoints of \( RS \) and \( TS \)? Wait, no. Wait, \( U \) is midpoint of \( RS \), \( V \) is midpoint of \( TS \), \( W \) is midpoint of \( RT \). Wait, actually, the side opposite to \( \overline{UV} \) would be \( \overline{RT} \)? Wait, no. Wait, let's see: \( U \) is midpoint of \( RS \), \( V \) is midpoint of \( TS \), so the segment \( UV \) should be parallel to \( RT \) and half its length. Wait, \( RT \) has length 10? Wait, no, the length of \( RT \) is 10? Wait, the diagram shows \( RT \) with length 10? Wait, the left side \( RT \) is labeled 10. So \( RT = 10 \). Then by Midline Theorem, \( UV=\frac{1}{2}RT \). Wait, no, wait: Wait, \( U \) is midpoint of \( RS \), \( V \) is midpoint of \( TS \), so the midline \( UV \) should be parallel to \( RT \) and half its length. So \( RT = 10 \), so \( UV=\frac{10}{2}=5 \)? Wait, no, maybe I got the sides wrong. Wait, let's re-examine. The triangle \( RST \): sides \( RT = 10 \), \( RS = 12 \), \( TS = 14 \). \( U \) is midpoint of \( RS \), \( V \) is midpoint of \( TS \), \( W \) is midpoint of \( RT \). Then the midline connecting midpoints of \( RS \) and \( TS \) (i.e., \( U \) and \( V \)) should be parallel to \( RT \) and half its length. So \( RT = 10 \), so \( UV=\frac{1}{2} \times RT=\frac{1}{2} \times 10 = 5 \)? Wait, but maybe I mixed up. Wait, another way: the midline theorem says that the segment connecting midpoints of two sides is half the length of the third side. So if \( U \) is midpoint of \( RS \) and \( V \) is midpoint of \( TS \), then the third side is \( RT \), so \( UV=\frac{1}{2}RT \). \( RT = 10 \), so \( UV = 5 \). Wait, but let's check again. Alternatively, maybe \( U \) is midpoint of \( RS \), \( V \) is midpoint of \( TS \), so the side \( RS = 12 \), \( TS = 14 \), \( RT = 10 \). So the midline \( UV \) connects midpoints of \( RS \) and \( TS \), so it's parallel to \( RT \) and half its length. So \( RT = 10 \), so \( UV = 5 \). Wait, but maybe I made a mistake. Wait, no, the midline theorem: in a triangle, the segment joining the midpoints of two sides is parallel to the third side and half as long. So if \( U \) is midpoint of \( RS \) and \( V \) is midpoint of \( TS \), then the two sides are \( RS \) and \( TS \), so the third side is \( RT \). Therefore, \( UV \parallel RT \) and \( UV=\frac{1}{2}RT \). Since \( RT = 10 \), then \( UV = \frac{10}{2}=5 \). Wait, but let's confirm. Alternatively, maybe \( U \) is midpoint of \( RS \), \( V \) is midpoint of \( TS \), so the length of \( RS \) is 12, \( TS \) is 14, \( RT \) is 10. So the midline \( UV \) should be half of \( RT \), so 5. So the length of \( UV \) is 5.

Answer:

\( 5 \)