Hello, foreverbrokenpromises!

I can't get your diagram to show up

. . but I think I've figured it out.

$\displaystyle PQRS$ is an isosceles trapezoid with: .$\displaystyle PS=QR=17,\;PQ = 16,\;SR = 32$

$\displaystyle SR$ is extended to $\displaystyle V$ so that $\displaystyle RV = 16.$

Point $\displaystyle Q$ is on the hypotenue $\displaystyle ST$ of right triangle $\displaystyle STV.$

The length of $\displaystyle QT$ is:

. . $\displaystyle (a)\;26\qquad (b)\;3\sqrt{89} \qquad (c)\;4\sqrt{57} \qquad (d)\;3\sqrt{93}\qquad (e)\;39$

So I've figured out that the height of the trapezoid is 15 . Right!

. . and that $\displaystyle QS \:=\: \sqrt{801} \:=\:3\sqrt{89}$ . Yes!

Can someone help me find $\displaystyle QT$ with this information given and explain why? Code:

o T
* |
* |
* |
* |
P 16 Q * |
o-----------o |
/: * :\ |
/ : * : \ |
/ : * : \ |
/ * 15: \ |
/ * : : \ |
o-----+-----------+-----o - - - - - o
S 8 A 16 B 8 R 16 V

Draw $\displaystyle PA \perp SR$

Draw $\displaystyle QB \perp SR$

Then: .$\displaystyle SA = 8,\;AB = 16,\;BR = 8$

We see that: .$\displaystyle \Delta QBS \sim \Delta TVS$

Hence: .$\displaystyle \frac{ST}{SQ} \:=\:\frac{SV}{SB} \quad\Rightarrow\quad \frac{ST}{SQ} \:=\:\frac{48}{24} \quad\Rightarrow\quad ST \:=\:2\!\cdot\!QS $

Divide by 2: . $\displaystyle \tfrac{1}{2}ST \:=\:QS$

Since $\displaystyle QT = \tfrac{1}{2}ST$, we have: .$\displaystyle QT \:=\:QS \:=\:3\sqrt{89} $ . . . answer (b)