Question

complete the statements to verify that the triangles are similar.
\\( \frac { q r } { t u } = \\)
\\( \frac { p r } { s u } = \\)
\\( \frac { p q } { s t } = \frac { \sqrt { 5 2 } } { \sqrt { 1 3 } } = \\)
therefore, \\( \triangle p q r \sim \triangle s t u \\) by the theorem.

Explanation:

Step1: Calculate \( QR \) and \( TU \)

\( QR=\vert4 - 0\vert = 4\), \( TU=\vert- 2-(-4)\vert=2\). Then \(\frac{QR}{TU}=\frac{4}{2}=2\)

Step2: Calculate \( PR \) and \( SU \)

\( PR=\vert4-(-2)\vert = 6\), \( SU=\vert2-(-1)\vert=3\). Then \(\frac{PR}{SU}=\frac{6}{3}=2\)

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

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

Step4: Determine the similarity theorem

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

Answer:

\(\frac{QR}{TU}=2\), \(\frac{PR}{SU}=2\), \(\frac{\sqrt{52}}{\sqrt{13}} = 2\), SSS (Side - Side - Side) similarity theorem