Question

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=27, and PR= 21, what is the value of QR

Answer 1

To solve the problem, let's establish the relationships based on the given information.

1. We have four collinear points: P, Q, R, and S.
2. The arrangement of the points is: P -- Q -- R -- S.
3. We're given that \( PQ = RS \).
4. The total distance \( PS = 27 \) and the distance \( PR = 21 \).
5. We want to find the value of \( QR \).

Now, let's define the distances based on the points:

- Let \( PQ = x \).
- Since \( PR = 21 \) and \( Q \) is between \( P \) and \( R \), we have:
\[
PR = PQ + QR = x + QR.
\]
We know that \( PR = 21 \), so:
\[
x + QR = 21. \tag{1}
\]

- Next, since \( R \) is between \( Q \) and \( S \), we can express \( RS \) as:
\[
RS = PS - PR.
\]
Using \( PS = 27 \) and \( PR = 21 \):
\[
RS = 27 - 21 = 6. \tag{2}
\]

- From the first point, \( PQ = RS \), so we have:
\[
x = 6. \tag{3}
\]

Now, substituting \( x = 6 \) back into equation (1):
\[
6 + QR = 21.
\]
Subtracting 6 from both sides gives:
\[
QR = 21 - 6 = 15.
\]

Thus, the value of \( QR \) is \(\boxed{15}\).