Question

use the graph of four triangles. complete the statement below to prove that \\( \triangle mno \\) is not congruent to \\( \triangle pqr \\). the segment \\( qr \\) cannot be mapped to \\( \square \overline{jk} \square \overline{no} \square \overline{mn} \square \overline{mo} \square \overline{su} \\) by any composition of rigid transformations.

Explanation:

Step1: Calculate the length of \( \overline{QR} \)

Use the distance formula \( d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2} \). For \( Q(1,5) \) and \( R(2,2) \), \( QR=\sqrt{(2 - 1)^2+(2 - 5)^2}=\sqrt{1 + 9}=\sqrt{10} \).

Step2: Calculate the length of \( \overline{NO} \)

For \( N(-5,1) \) and \( O(-4,4) \), \( NO=\sqrt{(-4+5)^2+(4 - 1)^2}=\sqrt{1+9}=\sqrt{10} \).

Step3: Calculate the length of \( \overline{MN} \)

For \( M(-2,7) \) and \( N(-5,1) \), \( MN=\sqrt{(-5 + 2)^2+(1 - 7)^2}=\sqrt{9 + 36}=\sqrt{45}=3\sqrt{5} \).

Step4: Calculate the length of \( \overline{MO} \)

For \( M(-2,7) \) and \( O(-4,4) \), \( MO=\sqrt{(-4 + 2)^2+(4 - 7)^2}=\sqrt{4+9}=\sqrt{13} \).

Step5: Calculate the length of \( \overline{SU} \)

For \( S(-1,-5) \) and \( U(3,-1) \), \( SU=\sqrt{(3 + 1)^2+(-1+5)^2}=\sqrt{16 + 16}=\sqrt{32}=4\sqrt{2} \).

Step6: Calculate the length of \( \overline{JK} \)

For \( J(-8,-2) \) and \( K(-3,-1) \), \( JK=\sqrt{(-3 + 8)^2+(-1 + 2)^2}=\sqrt{25+1}=\sqrt{26} \).

Since \( QR=\sqrt{10} \), \( NO=\sqrt{10} \), but \( MN = 3\sqrt{5}\), \(MO=\sqrt{13}\), \(SU = 4\sqrt{2}\), \(JK=\sqrt{26}\). Rigid transformations (translations, rotations, reflections) preserve length.

Answer:

\(\overline{MN}\)