Question

the graph shows quadrilaterals defg and mnop. is defg congruent to mnop? justify your answer. yes, because a translation left 10 units and down 12 units maps defg onto mnop. yes, because a rotation 180° around the origin maps defg onto mnop. no, because \\( \overline { fg } \\) and \\( \overline { op } \\) do not have the same length. no, because \\( \overline { de } \\) and \\( \overline { mn } \\) do not have the same length.

Explanation:

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

Using the distance formula \( d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2} \). For \( D(4,8) \) and \( E(1,6) \), \( x_1 = 1,y_1 = 6,x_2 = 4,y_2 = 8 \).
\( DE=\sqrt{(4 - 1)^2+(8 - 6)^2}=\sqrt{3^2+2^2}=\sqrt{9 + 4}=\sqrt{13} \)

Step2: Calculate the length of \( \overline{MN} \)

For \( M(- 4,-4) \) and \( N(-7,-6) \), \( x_1=-7,y_1=-6,x_2=-4,y_2=-4 \).
\( MN=\sqrt{(-4+7)^2+(-4 + 6)^2}=\sqrt{3^2+2^2}=\sqrt{9+4}=\sqrt{13} \)

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

For \( F(6,2) \) and \( G(7,5) \), \( x_1 = 6,y_1 = 2,x_2 = 7,y_2 = 5 \).
\( FG=\sqrt{(7 - 6)^2+(5 - 2)^2}=\sqrt{1^2+3^2}=\sqrt{1+9}=\sqrt{10} \)

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

For \( O(-5,-9) \) and \( P(-3,-7) \), \( x_1=-5,y_1=-9,x_2=-3,y_2=-7 \).
\( OP=\sqrt{(-3 + 5)^2+(-7+9)^2}=\sqrt{2^2+2^2}=\sqrt{4 + 4}=\sqrt{8}=2\sqrt{2} \)

Since \( FG=\sqrt{10}\) and \(OP = 2\sqrt{2}\approx2.83\), \(FG
eq OP\)

Answer:

No, because \( \overline{FG} \) and \( \overline{OP} \) do not have the same length.