Question

R is the midpoint of
QS
and T is the midpoint of
PS
.
If PQ=u+39 and RT=u–2, what is the value of u?
Q
P
T
S
R
u=

Answer 1

To find the value of \( u \), we can use the information about midpoints and the lengths given.

1. We know that \( R \) is the midpoint of \( QS \), so:
\[
QR = RS
\]
This means \( QR + RS = QS \), and since both segments are equal, \( QS = 2QR \).

2. Similarly, \( T \) is the midpoint of \( PS \), so:
\[
PT = TS
\]
This means \( PT + TS = PS \), and since both segments are equal, \( PS = 2PT \).

3. We are given the lengths:
\[
PQ = u + 39 \quad \text{and} \quad RT = u - 2
\]

4. Since \( R \) and \( T \) are midpoints, we can express \( QS \) and \( PS \) in terms of \( PQ \) and \( RT \):
- From point \( P \) to point \( Q \) to point \( S \) through midpoints:
- \( QS = PQ + PS = (u + 39) + PS \)
- From point \( R \) to point \( T \):
- \( RT = \frac{1}{2} QS \)
- \( RT = \frac{1}{2}(PQ + PS) = \frac{1}{2}((u + 39) + PS) \)

5. We can also express \( PS \) in terms of \( RT \):
\[
PS = 2RT = 2(u - 2) = 2u - 4
\]

6. Substitute \( PS \) back into the equation from steps 4 and 5:
\[
QS = u + 39 + PS = (u + 39) + (2u - 4) = 3u + 35
\]

7. Since \( QS = 2RT = 2(u - 2) = 2u - 4 \), we can set the two expressions for \( QS \) equal to each other:
\[
3u + 35 = 2u - 4
\]

8. Solve for \( u \):
\[
3u - 2u = -4 - 35
\]
\[
u = -39
\]

The value of \( u \) is \( \boxed{-39} \).