Question

what can be concluded about triangles rst and vtu?
the triangles are congruent and similar with a
proportional relationship of 1.
the triangles are congruent but not similar.
the triangles are similar but not congruent.
there is not enough information to determine if the
triangles are congruent or similar.

Explanation:

Step1: Calculate the side lengths of triangle \(RST\)

Using the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
For \(RT\): \(x_1=-4,y_1 = 4,x_2=0,y_2 = 0\), \(RT=\sqrt{(0 + 4)^2+(0 - 4)^2}=\sqrt{16 + 16}=\sqrt{32}=4\sqrt{2}\).
For \(ST\): \(x_1 = 4,y_1 = 4,x_2=0,y_2 = 0\), \(ST=\sqrt{(0 - 4)^2+(0 - 4)^2}=\sqrt{16+16}=\sqrt{32}=4\sqrt{2}\).
For \(RS\): \(x_1=-4,y_1 = 4,x_2 = 4,y_2 = 4\), \(RS=\sqrt{(4 + 4)^2+(4 - 4)^2}=\sqrt{64}=8\).

Step2: Calculate the side lengths of triangle \(VTU\)

For \(VT\): \(x_1=-4,y_1=-4,x_2=0,y_2 = 0\), \(VT=\sqrt{(0 + 4)^2+(0 + 4)^2}=\sqrt{16 + 16}=\sqrt{32}=4\sqrt{2}\).
For \(UT\): \(x_1 = 4,y_1=-4,x_2=0,y_2 = 0\), \(UT=\sqrt{(0 - 4)^2+(0 + 4)^2}=\sqrt{16+16}=\sqrt{32}=4\sqrt{2}\).
For \(VU\): \(x_1=-4,y_1=-4,x_2 = 4,y_2=-4\), \(VU=\sqrt{(4 + 4)^2+(-4 + 4)^2}=\sqrt{64}=8\).

Step3: Check congruence and similarity

Since \(RT = VT\), \(ST=UT\), \(RS = VU\), by SSS (Side - Side - Side) congruence criterion, \(\triangle RST\cong\triangle VTU\).
All congruent triangles are similar with a scale factor (proportional relationship) of \(1\) (because \(\frac{RT}{VT}=\frac{ST}{UT}=\frac{RS}{VU}=1\)).

Answer:

The triangles are congruent and similar with a proportional relationship of 1.