Question

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are \\( \triangle stu \\) and \\( \triangle def \\) congruent?

Explanation:

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

Use the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
For \(ST\): \(T(- 9,0)\), \(S(-6,6)\)
\(ST=\sqrt{(-6 + 9)^2+(6 - 0)^2}=\sqrt{9 + 36}=\sqrt{45}=3\sqrt{5}\)
For \(SU\): \(S(-6,6)\), \(U(2,0)\)
\(SU=\sqrt{(2 + 6)^2+(0 - 6)^2}=\sqrt{64 + 36}=\sqrt{100} = 10\)
For \(TU\): \(T(-9,0)\), \(U(2,0)\)
\(TU=\sqrt{(2 + 9)^2+(0 - 0)^2}=\sqrt{121}=11\)

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

For \(DE\): \(D(5,4)\), \(E(9,10)\)
\(DE=\sqrt{(9 - 5)^2+(10 - 4)^2}=\sqrt{16+36}=\sqrt{52}=2\sqrt{13}\)
For \(DF\): \(D(5,4)\), \(F(-1,10)\)
\(DF=\sqrt{(-1 - 5)^2+(10 - 4)^2}=\sqrt{36 + 36}=\sqrt{72}=6\sqrt{2}\)
For \(EF\): \(E(9,10)\), \(F(-1,10)\)
\(EF=\sqrt{(-1 - 9)^2+(10 - 10)^2}=\sqrt{100}=10\)

Step3: Compare the side lengths

Since \(ST
eq DE\), \(ST
eq DF\), \(SU = EF\) (but other sides are not equal), \(TU
eq DE\), \(TU
eq DF\)

Answer:

No, \(\triangle STU\) and \(\triangle DEF\) are not congruent.