Question

1. use the coordinates below to determine if \\( \triangle a b c \\) and \\( \triangle d e f \\) are congruent. \\( \triangle a b c: a(2,-8), b(-5,-2), c(-7,3) ; \triangle d e f: d(-9,7), e(-11,12), f(-2,1) \\) \\( a b= \\) \\( d e= \\) \\( b c= \\) \\( e f= \\) \\( a c= \\) \\( d f= \\) 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 \(B(-5,-2)\), \(x_1 = 2,y_1=-8,x_2=-5,y_2 = -2\).

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Step2: Calculate the length of \(DE\)

For \(D(-9,7)\) and \(E(-11,12)\), \(x_1=-9,y_1 = 7,x_2=-11,y_2=12\).

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Step3: Calculate the length of \(BC\)

For \(B(-5,-2)\) and \(C(-7,3)\), \(x_1=-5,y_1=-2,x_2=-7,y_2 = 3\).

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Step4: Calculate the length of \(EF\)

For \(E(-11,12)\) and \(F(-2,1)\), \(x_1=-11,y_1=12,x_2=-2,y_2 = 1\).

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Step5: Calculate the length of \(AC\)

For \(A(2,-8)\) and \(C(-7,3)\), \(x_1=2,y_1=-8,x_2=-7,y_2 = 3\).

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Step6: Calculate the length of \(DF\)

For \(D(-9,7)\) and \(F(-2,1)\), \(x_1=-9,y_1=7,x_2=-2,y_2 = 1\).

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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:

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