Question

a diagram with two right triangles is shown.
prove that $\triangle pqs \sim \triangle prt$.
given:
$pq = 1.5$, $qr = 4.5$
$sq = 1$, $tr = ?$
$\angle pqs$ and $\angle prt$ are right angles (given)
$pr = 6$ (segment addition)
$\frac{pq}{pr} = \frac{sq}{tr}$ (corresponding sides are proportional)
$\triangle pqs \sim \triangle prt$ ( ? )
$\angle pqs \cong \angle prt$ (all right angles are congruent)

Explanation:

Step1: Calculate PR length

$PR = PQ + QR = 1.5 + 4.5 = 6$

Step2: Find similarity ratio

$\frac{PQ}{PR} = \frac{1.5}{6} = \frac{1}{4}$

Step3: Solve for TR length

$\frac{SQ}{TR} = \frac{1}{4} \implies TR = 4 \times SQ = 4 \times 1 = 4$

Step4: Verify right angles congruence

$\angle PQS \cong \angle PRT$ (both are right angles, so congruent)

Step5: Confirm proportional sides

$\frac{PQ}{PR} = \frac{SQ}{TR} = \frac{1}{4}$ (corresponding sides proportional)

Step6: Apply SAS similarity criterion

Two triangles with a congruent included angle and proportional surrounding sides are similar.

Answer:

$\triangle PQS \sim \triangle PRT$ by the Side-Angle-Side (SAS) Similarity Criterion. The missing value for $TR$ is 4.