Question

the graph shows quadrilaterals defg and mnop. is defg congruent to mnop? justify your answer.

Explanation:

Step1: Calculate the side lengths of DEFG

Use the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
For \(DE\): \(D(4,8)\), \(E(1,6)\), \(d_{DE}=\sqrt{(4 - 1)^2+(8 - 6)^2}=\sqrt{9 + 4}=\sqrt{13}\).
For \(EF\): \(E(1,6)\), \(F(6,2)\), \(d_{EF}=\sqrt{(6 - 1)^2+(2 - 6)^2}=\sqrt{25+16}=\sqrt{41}\).
For \(FG\): \(F(6,2)\), \(G(7,5)\), \(d_{FG}=\sqrt{(7 - 6)^2+(5 - 2)^2}=\sqrt{1 + 9}=\sqrt{10}\).
For \(GD\): \(G(7,5)\), \(D(4,8)\), \(d_{GD}=\sqrt{(4 - 7)^2+(8 - 5)^2}=\sqrt{9+9}=\sqrt{18}\).

Step2: Calculate the side lengths of MNOP

For \(MN\): \(M(-5,-4)\), \(N(-8,-6)\), \(d_{MN}=\sqrt{(-5+8)^2+(-4 + 6)^2}=\sqrt{9+4}=\sqrt{13}\).
For \(NO\): \(N(-8,-6)\), \(O(-6,-10)\), \(d_{NO}=\sqrt{(-6 + 8)^2+(-10+6)^2}=\sqrt{4 + 16}=\sqrt{20}
eq\sqrt{41}\).

Answer:

No, DEFG is not congruent to MNOP because the side lengths are not all equal. For example, \(EF=\sqrt{41}\) and \(NO=\sqrt{20}\), and \(\sqrt{41}
eq\sqrt{20}\).