Question

complete the statements to verify that the triangles are similar. \\(\frac{qr}{tu} = \square\\) \\(\frac{pr}{su} = \square\\) \\(\frac{pq}{st} = \frac{\sqrt{52}}{\sqrt{13}} = \square\\) therefore, \\(\triangle pqr \sim \triangle stu\\) by the \\(\square\\) theorem. points: r(-2, 4), p(4, 4), q(-2, 0), t(-1, -2), u(-1, -4), s(2, -4) with coordinate grid.

Explanation:

Step1: Calculate QR and TU

QR: vertical distance between Q(-2,0) & R(-2,4): $4-0=4$. TU: vertical distance between T(-1,-2) & U(-1,-4): $-2 - (-4)=2$. Ratio: $\frac{4}{2}=2$.

Step2: Calculate PR and SU

PR: horizontal distance between P(4,4) & R(-2,4): $4 - (-2)=6$. SU: horizontal distance between S(2,-4) & U(-1,-4): $2 - (-1)=3$. Ratio: $\frac{6}{3}=2$.

Step3: Simplify $\frac{\sqrt{52}}{\sqrt{13}}$

$\sqrt{52}=2\sqrt{13}$, so $\frac{2\sqrt{13}}{\sqrt{13}}=2$.

Step4: Identify similarity theorem

All corresponding sides have equal ratio (2:1), so SSS Similarity.

Answer:

2; 2; 2; SSS Similarity