Question

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

Explanation:

Step1: Calculate QR and TU

QR is the vertical distance between Q(-2,0) and R(-2,4). So \( QR = |4 - 0| = 4 \).
TU is the vertical distance between T(-1,-2) and U(-1,-4). So \( TU = |-4 - (-2)| = |-2| = 2 \). Then \( \frac{QR}{TU} = \frac{4}{2} = 2 \).

Step2: Calculate PR and SU

PR is the horizontal distance between P(4,4) and R(-2,4). So \( PR = |4 - (-2)| = 6 \).
SU is the horizontal distance between S(2,-4) and U(-1,-4). So \( SU = |2 - (-1)| = 3 \). Then \( \frac{PR}{SU} = \frac{6}{3} = 2 \).

Step3: Simplify \( \frac{\sqrt{52}}{\sqrt{13}} \)

Using the property \( \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \), we have \( \frac{\sqrt{52}}{\sqrt{13}} = \sqrt{\frac{52}{13}} = \sqrt{4} = 2 \).

Step4: Determine the similarity theorem

Since the ratios of all corresponding sides (\( \frac{QR}{TU} = \frac{PR}{SU} = \frac{PQ}{ST} = 2 \)) are equal, the triangles are similar by the SSS (Side - Side - Side) similarity theorem.

Answer:

\( \frac{QR}{TU}=\boldsymbol{2} \), \( \frac{PR}{SU}=\boldsymbol{2} \), \( \frac{PQ}{ST}=\boldsymbol{2} \), and \( \triangle PQR \sim \triangle STU \) by the \(\boldsymbol{SSS\ similarity}\) theorem.