Question

verifying that triangles are similar
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 with options 1/4, 1/2, 2, 4
\\(\frac{pq}{st}\\) = dropdown
therefor \\(\triangle pqr \sim \triangle stu\\) by the dropdown theorem.

Explanation:

Step1: Calculate QR length

QR is vertical from Q(-2,0) to R(-2,4). Length: \( |4 - 0| = 4 \).

Step2: Calculate TU length

TU is vertical from T(-1,-2) to U(-1,-4). Length: \( |-4 - (-2)| = 2 \).

Step3: Find \(\frac{QR}{TU}\)

\(\frac{QR}{TU} = \frac{4}{2} = 2\).

Step4: Calculate PR length

PR is horizontal from R(-2,4) to P(4,4). Length: \( |4 - (-2)| = 6 \).

Step5: Calculate SU length

SU is horizontal from S(2,-4) to U(-1,-4). Length: \( |2 - (-1)| = 3 \).

Step6: Find \(\frac{PR}{SU}\)

\(\frac{PR}{SU} = \frac{6}{3} = 2\).

Step7: Calculate PQ length

Using distance formula: \( PQ = \sqrt{(4 - (-2))^2 + (4 - 0)^2} = \sqrt{36 + 16} = \sqrt{52} = 2\sqrt{13} \).

Step8: Calculate ST length

Using distance formula: \( ST = \sqrt{(2 - (-1))^2 + (-4 - (-2))^2} = \sqrt{9 + 4} = \sqrt{13} \).

Step9: Find \(\frac{PQ}{ST}\)

\(\frac{PQ}{ST} = \frac{2\sqrt{13}}{\sqrt{13}} = 2\).

Step10: Determine similarity theorem

Since all corresponding sides are in ratio 2 (SSS similarity), the triangles are similar by SSS (Side - Side - Side) similarity theorem.

Answer:

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