Question

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in △qrs, rx = 48 and qw = 30. find each length.

1. rw

1. wx

1. qz

1. wz

(there is a triangle diagram on the right with points r, q, s, and segments with markings indicating midpoints or angle bisectors, and point w inside the triangle.)

Explanation:

Step1: Identify Centroid Properties

In a triangle, the centroid (point \( W \)) divides each median into a ratio of \( 2:1 \), with the longer segment being closer to the vertex. Also, the centroid is the intersection of the medians, so each median is split into two parts where the distance from the vertex to the centroid is \( \frac{2}{3} \) of the median, and from centroid to midpoint is \( \frac{1}{3} \) of the median.

Step2: Find \( RW \) (Part 1)

For median \( RX \) (since \( X \) is the midpoint of \( QS \), as \( QX = XS \) from the markings), the centroid \( W \) divides \( RX \) such that \( RW=\frac{2}{3}RX \). Given \( RX = 48 \), so \( RW=\frac{2}{3}\times48 = 32 \).

Step3: Find \( WX \) (Part 2)

Since \( RX = RW + WX \) and \( RW=\frac{2}{3}RX \), then \( WX=\frac{1}{3}RX \). So \( WX=\frac{1}{3}\times48 = 16 \).

Step4: Find \( QZ \) (Part 3)

For median \( QZ \) (since \( Y \) is the midpoint of \( RQ \), as \( RY = YQ \) from the markings), the centroid \( W \) divides \( QZ \) such that \( QW=\frac{2}{3}QZ \). Given \( QW = 30 \), we solve for \( QZ \): \( QZ=\frac{3}{2}\times QW=\frac{3}{2}\times30 = 45 \).

Step5: Find \( WZ \) (Part 4)

Since \( QZ = QW + WZ \) and \( QW=\frac{2}{3}QZ \), then \( WZ=\frac{1}{3}QZ \). We know \( QZ = 45 \), so \( WZ=\frac{1}{3}\times45 = 15 \).

Answer:

1. \( RW = 32 \)
2. \( WX = 16 \)
3. \( QZ = 45 \)
4. \( WZ = 15 \)