Question

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are \\( \triangle tuv \\) and \\( \triangle fgh \\) congruent?

Explanation:

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

Use the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
For \(TU\): \(T(- 10,8)\), \(U(-1,8)\), \(d_{TU}=\sqrt{(-1+10)^2+(8 - 8)^2}=\sqrt{81+0}=9\).
For \(TV\): \(T(-10,8)\), \(V(-10,-1)\), \(d_{TV}=\sqrt{(-10 + 10)^2+(-1 - 8)^2}=\sqrt{0 + 81}=9\).
For \(UV\): \(U(-1,8)\), \(V(-10,-1)\), \(d_{UV}=\sqrt{(-10 + 1)^2+(-1 - 8)^2}=\sqrt{81+81}=\sqrt{162}=9\sqrt{2}\).

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

For \(FH\): \(F(8,1)\), \(H(-1,1)\), \(d_{FH}=\sqrt{(8 + 1)^2+(1 - 1)^2}=\sqrt{81+0}=9\).
For \(FG\): \(F(8,1)\), \(G(8,-7)\), \(d_{FG}=\sqrt{(8 - 8)^2+(-7 - 1)^2}=\sqrt{0 + 64}=8\).
For \(GH\): \(G(8,-7)\), \(H(-1,1)\), \(d_{GH}=\sqrt{(8 + 1)^2+(-7 - 1)^2}=\sqrt{81+64}=\sqrt{145}\).

Since the side - lengths of \(\triangle TUV\) (\(9,9,9\sqrt{2}\)) and \(\triangle FGH\) (\(9,8,\sqrt{145}\)) are not equal.

Answer:

\(\triangle TUV\) and \(\triangle FGH\) are not congruent.