Question

triangle qrs is formed by connecting the midpoints of the side of triangle nop. the lengths of the sides of triangle nop are shown. what is the length of overline{qs}? figures not necessarily drawn to scale.

Explanation:

Step1: Identify Midsegment Theorem

The Midsegment Theorem (or Midline Theorem) states that the segment connecting the midpoints of two sides of a triangle is parallel to the third side and half as long. Here, \( Q \), \( R \), \( S \) are midpoints of \( NO \), \( OP \), \( PN \) respectively (since \( QRS \) is formed by connecting midpoints of \( NOP \)). So, \( QS \) is a midsegment of \( \triangle NOP \), and it should be half the length of the third side (here, \( OP \) has length 6? Wait, no, looking at the diagram: sides of \( \triangle NOP \) are all 6? Wait, the labels: \( NO \) has length 6 (from \( N \) to \( O \) with \( Q \) as midpoint), \( OP \) has length 6 (from \( O \) to \( P \) with \( R \) as midpoint), \( PN \) has length 6 (from \( P \) to \( N \) with \( S \) as midpoint). Wait, actually, \( Q \) is midpoint of \( NO \), \( S \) is midpoint of \( PN \), so \( QS \) connects midpoints of \( NO \) and \( PN \), so the third side is \( OP \), which has length 6. Wait, no, let's re-express: in \( \triangle NOP \), \( Q \) is midpoint of \( NO \), \( S \) is midpoint of \( NP \), so \( QS \) is midsegment parallel to \( OP \) and \( QS=\frac{1}{2}OP \). Since \( OP = 6 \) (from the diagram, \( OP \) is labeled with length 6? Wait, the diagram shows \( OP \) with length 6? Wait, the side from \( O \) to \( P \) (via \( R \)) is labeled 6? Wait, the text says "the lengths of the sides of triangle NOP are shown" and the diagram has \( NO \), \( OP \), \( PN \) each with length 6? Wait, maybe \( \triangle NOP \) is equilateral with side length 6. Then, \( Q \) is midpoint of \( NO \), \( S \) is midpoint of \( PN \), so \( QS \) is midsegment, so \( QS=\frac{1}{2}OP \). Since \( OP = 6 \), then \( QS=\frac{6}{2}=3 \).

Step2: Apply Midsegment Theorem

Using the Midsegment Theorem: \( QS=\frac{1}{2} \times \text{length of } OP \). From the diagram, \( OP = 6 \) (since the side \( OP \) has length 6, as marked). Therefore, \( QS=\frac{1}{2} \times 6 = 3 \).

Answer:

\( 3 \)