Question

1. use the coordinates below to determine if △abc and △def are congruent. △abc: a(2, -8), b(-5, -2), c(-7, 3); △def: d(-9, 7), e(-11, 12), f(-2, 1) ab = bc = ac = de = ef = df = are the triangles congruent? if yes, explain your reasoning and write a congruency statement.

Explanation:

Step1: Calculate the length of \(AB\)

Use the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). For \(A(2,-8)\) and \(B(-5,-2)\), \(x_1 = 2,y_1=-8,x_2=-5,y_2 = -2\).
\(AB=\sqrt{(-5 - 2)^2+(-2+8)^2}=\sqrt{(-7)^2+6^2}=\sqrt{49 + 36}=\sqrt{85}\)

Step2: Calculate the length of \(BC\)

For \(B(-5,-2)\) and \(C(-7,3)\), \(x_1=-5,y_1=-2,x_2=-7,y_2 = 3\).
\(BC=\sqrt{(-7 + 5)^2+(3 + 2)^2}=\sqrt{(-2)^2+5^2}=\sqrt{4+25}=\sqrt{29}\)

Step3: Calculate the length of \(AC\)

For \(A(2,-8)\) and \(C(-7,3)\), \(x_1 = 2,y_1=-8,x_2=-7,y_2 = 3\).
\(AC=\sqrt{(-7 - 2)^2+(3 + 8)^2}=\sqrt{(-9)^2+11^2}=\sqrt{81+121}=\sqrt{202}\)

Step4: Calculate the length of \(DE\)

For \(D(-9,7)\) and \(E(-11,12)\), \(x_1=-9,y_1 = 7,x_2=-11,y_2 = 12\).
\(DE=\sqrt{(-11 + 9)^2+(12 - 7)^2}=\sqrt{(-2)^2+5^2}=\sqrt{4 + 25}=\sqrt{29}\)

Step5: Calculate the length of \(EF\)

For \(E(-11,12)\) and \(F(-2,1)\), \(x_1=-11,y_1 = 12,x_2=-2,y_2 = 1\).
\(EF=\sqrt{(-2 + 11)^2+(1 - 12)^2}=\sqrt{9^2+(-11)^2}=\sqrt{81+121}=\sqrt{202}\)

Step6: Calculate the length of \(DF\)

For \(D(-9,7)\) and \(F(-2,1)\), \(x_1=-9,y_1 = 7,x_2=-2,y_2 = 1\).
\(DF=\sqrt{(-2 + 9)^2+(1 - 7)^2}=\sqrt{7^2+(-6)^2}=\sqrt{49+36}=\sqrt{85}\)

Step7: Check congruency

Since \(AB = DF=\sqrt{85}\), \(BC=DE=\sqrt{29}\), \(AC = EF=\sqrt{202}\), by SSS (Side - Side - Side) congruence criterion, \(\triangle ABC\cong\triangle DFE\)

Answer:

\(AB=\sqrt{85}\), \(BC=\sqrt{29}\), \(AC=\sqrt{202}\), \(DE=\sqrt{29}\), \(EF=\sqrt{202}\), \(DF=\sqrt{85}\). The triangles \(\triangle ABC\) and \(\triangle DFE\) are congruent by SSS congruence criterion. So, \(\triangle ABC\cong\triangle DFE\)