Question

In the diagram below, start overline, T, U, end overline
TU
is parallel to start overline, Q, R, end overline
QR
​
. If Q, R, equals, 15QR=15, T, U, equals, 10TU=10, and Q, S, equals, 9QS=9, find the length of start overline, T, S, end overline
TS
. Figures are not necessarily drawn to scale.
Q
R
S
T
U
Answer

Answer 1

Interpret the figure as a trapezoid with bases QR = 15 and TU = 10 and diagonals QU and RT intersecting at S.

Put coordinates Q(0,h), R(15,h), T(0,0), U(10,0). The intersection parameter t along QU (measured from Q) is
t = QR/(QR+TU) = 15/(15+10) = 3/5.

Thus
QS = t·|QU| = (3/5)·sqrt(10^2 + h^2) = 9,
so sqrt(100 + h^2) = 9·5/3 = 15 ⇒ h^2 = 125.

Now TS is the portion of RT from T to S:
TS = (1 − t)·|RT| = (2/5)·sqrt(15^2 + h^2) = (2/5)·sqrt(225 + 125)
= (2/5)·sqrt(350) = 2·sqrt(14).

Answer: TS = 2√14.