Question
RT
is a midsegment of △QSU.
If QU=p–2 and RT=p–31, what is the value of p?
U
Q
R
S
T
p=
Answers
GPT-4o mini
In triangle \( \triangle QSU \), the midsegment \( RT \) connects the midpoints of sides \( QS \) and \( QU \). According to the properties of midsegments in triangles, the length of the midsegment is half the length of the side it is parallel to.
In this case, \( RT \) is parallel to side \( QU \), and thus we have:
\[
RT = \frac{1}{2} QU
\]
Substituting the given lengths of \( QU \) and \( RT \):
\[
p - 31 = \frac{1}{2}(p - 2)
\]
To eliminate the fraction, multiply both sides by 2:
\[
2(p - 31) = p - 2
\]
Now, distribute on the left-hand side:
\[
2p - 62 = p - 2
\]
Next, isolate \( p \) by subtracting \( p \) from both sides:
\[
2p - p - 62 = -2
\]
This simplifies to:
\[
p - 62 = -2
\]
Now, add 62 to both sides:
\[
p = 60
\]
Therefore, the value of \( p \) is:
\[
\boxed{60}
\]
In this case, \( RT \) is parallel to side \( QU \), and thus we have:
\[
RT = \frac{1}{2} QU
\]
Substituting the given lengths of \( QU \) and \( RT \):
\[
p - 31 = \frac{1}{2}(p - 2)
\]
To eliminate the fraction, multiply both sides by 2:
\[
2(p - 31) = p - 2
\]
Now, distribute on the left-hand side:
\[
2p - 62 = p - 2
\]
Next, isolate \( p \) by subtracting \( p \) from both sides:
\[
2p - p - 62 = -2
\]
This simplifies to:
\[
p - 62 = -2
\]
Now, add 62 to both sides:
\[
p = 60
\]
Therefore, the value of \( p \) is:
\[
\boxed{60}
\]