Question

what is the length of (overline{sr})?
9 units
12 units
15 units
18 units

(diagram: right triangle (rqs) with right angle at (r), (rq = 20), (qs = 16), point (t) on (qs) with (rt) drawn)

Explanation:

Step1: Identify similar triangles

$\triangle STR \sim \triangle SRQ$ (right triangles share $\angle S$)

Step2: Set up proportion

$\frac{SR}{SQ} = \frac{ST}{SR}$
Given $SQ = ST + TQ = SR + 16$? No, correct: $SQ = ST + TQ$, let $SR = x$, $ST = SQ - 16 = \sqrt{x^2 + 20^2} - 16$. Use geometric mean: $SR^2 = ST \times SQ$
Let $SQ = y$, so $ST = y - 16$. Then $x^2 = (y - 16)y$, and $x^2 + 20^2 = y^2$

Step3: Substitute and solve

Substitute $x^2 = y^2 -16y$ into $x^2 + 400 = y^2$:
$y^2 -16y + 400 = y^2$
$-16y = -400$
$y = 25$

Step4: Calculate SR

$x^2 + 20^2 = 25^2$
$x^2 = 625 - 400 = 225$
$x = 15$

Answer:

15 units