Question

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are △ghi and △def congruent?

Explanation:

Step1: Calculate the side lengths of \(\triangle GHI\)

Using the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) or counting units on the grid.
For \(GH\): \(G(0,0)\), \(H(- 10,10)\). \(GH=\sqrt{(-10 - 0)^2+(10 - 0)^2}=\sqrt{100 + 100}=\sqrt{200}=10\sqrt{2}\)
For \(GI\): \(G(0,0)\), \(I(0,10)\). \(GI = 10\)
For \(HI\): \(H(-10,10)\), \(I(0,10)\). \(HI=\sqrt{(0+10)^2+(10 - 10)^2}=10\)

Step2: Calculate the side lengths of \(\triangle DEF\)

For \(DE\): \(D(0,-10)\), \(E(10,0)\). \(DE=\sqrt{(10 - 0)^2+(0 + 10)^2}=\sqrt{100+100}=\sqrt{200}=10\sqrt{2}\)
For \(DF\): \(D(0,-10)\), \(F(10,-10)\). \(DF = 10\)
For \(EF\): \(E(10,0)\), \(F(10,-10)\). \(EF=\sqrt{(10 - 10)^2+(-10 - 0)^2}=10\)

Step3: Check the congruence

Since \(GH = DE = 10\sqrt{2}\), \(GI=DF = 10\), \(HI = EF=10\) (by SSS - Side - Side - Side congruence criterion)

Answer:

Yes, \(\triangle GHI\) and \(\triangle DEF\) are congruent.