Question

in the figure below, points s, t, and u are the midpoints of the sides of △pqr. suppose tu = 24, pq = 36, and qr = 60. find the following lengths. pr = rs = su =

Explanation:

Step1: Apply Midline Theorem (Triangle Midsegment Theorem)

The 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. For \( \triangle PQR \), since \( T \) and \( U \) are midpoints of \( PQ \) and \( QR \) respectively, \( TU \parallel PR \) and \( TU=\frac{1}{2}PR \). We know \( TU = 24 \), so:

$$ PR = 2\times TU = 2\times24 = 48 $$

Step2: Find \( RS \)

Since \( S \) is the midpoint of \( PR \) (as \( S, T, U \) are midpoints of the sides), \( RS=\frac{1}{2}PR \). We found \( PR = 48 \), so:

$$ RS=\frac{1}{2}\times48 = 24 $$

Step3: Find \( SU \)

Since \( S \) and \( U \) are midpoints of \( PR \) and \( QR \) respectively, by the Midline Theorem, \( SU \parallel PQ \) and \( SU=\frac{1}{2}PQ \). We know \( PQ = 36 \), so:

$$ SU=\frac{1}{2}\times36 = 18 $$

Answer:

\( PR = 48 \)
\( RS = 24 \)
\( SU = 18 \)