Questions: Points P, Q, R, and S are collinear. Point Q is between P and R, R is between Q and S, and PQ=RS. If PS=29 and PR=21, what is the value of QR?

Solution Steps

To solve this problem, we need to use the given information about the distances between the points and set up an equation to find the unknown distance \( QR \).

1. We know that \( PQ = RS \) and \( PS = 29 \).
2. We also know that \( PR = 21 \).
3. Since \( Q \) is between \( P \) and \( R \), and \( R \) is between \( Q \) and \( S \), we can express \( PS \) as the sum of the segments \( PQ + QR + RS \).
4. Using the given distances, we can set up an equation to solve for \( QR \).

We are given four collinear points \( P, Q, R, \) and \( S \) with the following conditions:

• \( Q \) is between \( P \) and \( R \)
• \( R \) is between \( Q \) and \( S \)
• \( PQ = RS \)
• \( PS = 29 \)
• \( PR = 21 \)

We need to find the value of \( QR \).

Since \( Q \) is between \( P \) and \( R \), and \( R \) is between \( Q \) and \( S \), we can write the distances as: \[ PS = PQ + QR + RS \]

Given \( PS = 29 \) and \( PR = 21 \), we can also write: \[ PR = PQ + QR \]

We know \( PQ = RS \). Let \( PQ = RS = x \).

From the equation \( PR = PQ + QR \): \[ 21 = x + QR \] \[ QR = 21 - x \]

From the equation \( PS = PQ + QR + RS \): \[ 29 = x + (21 - x) + x \] \[ 29 = 21 + x \] \[ x = 29 - 21 \] \[ x = 8 \]

Now that we have \( x = 8 \): \[ QR = 21 - x \] \[ QR = 21 - 8 \] \[ QR = 13 \]

Final Answer