Question

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are \\( \triangle ghi \\) and \\( \triangle abc \\) congruent?

Explanation:

Step1: Calculate side lengths using distance formula

The distance formula is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
For \(\triangle ABC\):

• \(AB\): \(A(3,0)\), \(B(3,9)\), \(AB=\sqrt{(3 - 3)^2+(9 - 0)^2}=9\)
• \(AC\): \(A(3,0)\), \(C(10,5)\), \(AC=\sqrt{(10 - 3)^2+(5 - 0)^2}=\sqrt{49 + 25}=\sqrt{74}\)
• \(BC\): \(B(3,9)\), \(C(10,5)\), \(BC=\sqrt{(10 - 3)^2+(5 - 9)^2}=\sqrt{49+16}=\sqrt{65}\)

For \(\triangle GHI\):

• \(GH\): \(G(1,-9)\), \(H(-9,-9)\), \(GH=\sqrt{(-9 - 1)^2+(-9+9)^2}=10\)
• \(GI\): \(G(1,-9)\), \(I(-4,-3)\), \(GI=\sqrt{(-4 - 1)^2+(-3 + 9)^2}=\sqrt{25 + 36}=\sqrt{61}\)
• \(HI\): \(H(-9,-9)\), \(I(-4,-3)\), \(HI=\sqrt{(-4 + 9)^2+(-3 + 9)^2}=\sqrt{25+36}=\sqrt{61}\)

Step2: Compare side lengths

Since the side lengths of \(\triangle ABC\) (\(9,\sqrt{65},\sqrt{74}\)) and \(\triangle GHI\) (\(10,\sqrt{61},\sqrt{61}\)) are not equal.

Answer:

No, \(\triangle GHI\) and \(\triangle ABC\) are not congruent.