Question

triangles ptq, qsr, and pur, shown on the graph below, are similar right triangles.
solana writes 5 proportions to represent the slopes of the hypotenuses of the similar triangles along line rp.
determine if each of her proportions correctly represents the slopes or not.
drag each of solanas proportions to the correct box.
correctly represents slopes does not correctly represent slopes
::\\( \frac { p t } { u r } = \frac { q s } { s r } \\) ::\\( \frac { t u } { u s } = \frac { q s } { s r } \\) ::\\( \frac { p t } { t q } = \frac { q s } { s r } \\) ::\\( \frac { p u } { u r } = \frac { p t } { t q } \\) ::\\( \frac { q s } { s r } = \frac { p u } { u s } \\)

Explanation:

Step1: Find coordinates

Find coordinates of points: \(P(1,2)\), \(T(1,4)\), \(Q(3,4)\), \(S(4,7)\), \(R(6,7)\), \(U(1,7)\).

Step2: Calculate lengths

Calculate lengths: \(PT = 4 - 2=2\), \(TQ=3 - 1 = 2\), \(QS=7 - 4 = 3\), \(SR=6 - 4 = 2\), \(UR=6 - 1 = 5\), \(US=4 - 1 = 3\).

Step3: Check \(\frac{PT}{TQ}=\frac{QS}{SR}\)

\(\frac{PT}{TQ}=\frac{2}{2} = 1\), \(\frac{QS}{SR}=\frac{3}{3}=1\). So, \(\frac{PT}{TQ}=\frac{QS}{SR}\) is correct.

Step4: Check \(\frac{QS}{SR}=\frac{PU}{US}\)

\(PU = 7 - 2=5\), \(\frac{QS}{SR}=\frac{3}{3}=1\), \(\frac{PU}{US}=\frac{5}{3}
eq1\). So, \(\frac{QS}{SR}=\frac{PU}{US}\) is incorrect.

Step5: Check \(\frac{PT}{UR}=\frac{QS}{SR}\)

\(\frac{PT}{UR}=\frac{2}{5}\), \(\frac{QS}{SR}=1\). So, \(\frac{PT}{UR}=\frac{QS}{SR}\) is incorrect.

Step6: Check \(\frac{PU}{UR}=\frac{PT}{TQ}\)

\(\frac{PU}{UR}=\frac{5}{5} = 1\), \(\frac{PT}{TQ}=1\). So, \(\frac{PU}{UR}=\frac{PT}{TQ}\) is correct.

Step7: Check \(\frac{TU}{US}=\frac{QS}{SR}\)

\(TU = 7 - 4 = 3\), \(\frac{TU}{US}=\frac{3}{3}=1\), \(\frac{QS}{SR}=1\). So, \(\frac{TU}{US}=\frac{QS}{SR}\) is correct.

Answer:

Correctly Represents Slopes: \(\frac{PT}{TQ}=\frac{QS}{SR}\), \(\frac{PU}{UR}=\frac{PT}{TQ}\), \(\frac{TU}{US}=\frac{QS}{SR}\)
Does Not Correctly Represent Slopes: \(\frac{PT}{UR}=\frac{QS}{SR}\), \(\frac{QS}{SR}=\frac{PU}{US}\)