Question

triangle stu is formed by connecting the midpoints of the side of triangle pqr. the lengths of the sides of triangle pqr are shown. what is the length of ut? figures not necessarily drawn to scale.

Explanation:

Step1: Identify Midsegment Theorem

In a triangle, the segment connecting the midpoints of two sides (midsegment) is parallel to the third side and half its length. Here, \( S, T, U \) are midpoints, so \( UT \) is a midsegment.

Step2: Determine the Third Side Length

The side \( PQ \) has length \( 14 \) (since \( PS = SQ = 14/2 = 7 \), but we need the side parallel to \( UT \)). Wait, actually, \( T \) and \( U \) are midpoints of \( QR \) and \( PR \)? Wait, no, let's check the triangle. Wait, \( STU \) is formed by midpoints of \( PQR \). So \( UT \) should be parallel to \( PQ \) and half its length? Wait, no, maybe \( UT \) is parallel to \( PQ \)? Wait, \( PQ \) length is \( 14 \)? Wait, no, the sides: \( PQ \) is from \( P \) to \( Q \), with \( S \) as midpoint (since \( PS = SQ = 7 \), so \( PQ = 14 \)). Then \( UT \) is a midsegment, so \( UT=\frac{1}{2}PQ \)? Wait, no, maybe \( UT \) is parallel to \( PQ \), so \( UT = \frac{1}{2}PQ \). Wait, \( PQ = 14 \), so \( UT = 7 \)? Wait, no, maybe I mixed up. Wait, the side \( QR \) is \( 10 \)? Wait, no, the problem says "the lengths of the sides of triangle \( PQR \) are shown". Let's re-express: \( S, T, U \) are midpoints. So by Midsegment Theorem, the midsegment is half the length of the third side. So if \( UT \) is a midsegment, then it's half the length of the side it's parallel to. Let's see: \( S \) is midpoint of \( PQ \), \( T \) midpoint of \( QR \), \( U \) midpoint of \( PR \). Then \( UT \) connects midpoints of \( PR \) and \( QR \), so it's parallel to \( PQ \) and \( UT=\frac{1}{2}PQ \). \( PQ \) length: from the diagram, \( PS = SQ = 7 \), so \( PQ = 14 \). Thus, \( UT=\frac{1}{2}\times14 = 7 \)? Wait, no, maybe I made a mistake. Wait, the side \( QR \) is \( 10 \)? Wait, no, the problem says "the lengths of the sides of triangle \( PQR \) are shown". Let's check the labels: \( PQ \) has length \( 14 \) (since \( PS = SQ = 7 \)), \( QR \) has length \( 10 \) (since \( QT = TR = 5 \)? Wait, no, the diagram shows \( QT = 10 \)? Wait, no, the diagram: \( Q \) to \( T \) is labeled \( 10 \)? Wait, no, the user's diagram: \( Q \) to \( T \) is \( 10 \), \( P \) to \( U \) is \( 14 \), \( P \) to \( S \) is \( 14 \). Wait, maybe \( PQ = 14 \), and \( UT \) is midsegment, so \( UT = \frac{1}{2}PQ = 7 \). Wait, but let's confirm. Midsegment Theorem: In \( \triangle PQR \), if \( U \) is midpoint of \( PR \) and \( T \) is midpoint of \( QR \), then \( UT \parallel PQ \) and \( UT = \frac{1}{2}PQ \). Since \( PQ = 14 \), then \( UT = 7 \).

Answer:

\( 7 \)