Question

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are \\( \triangle i j k \\) and \\( \triangle r s t \\) congruent?

Explanation:

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

Use the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).

• For \(IJ\): \(I(-2,-8)\), \(J(-9,-9)\)

\(IJ=\sqrt{(-2 + 9)^2+(-8 + 9)^2}=\sqrt{49 + 1}=\sqrt{50}\)

• For \(JK\): \(J(-9,-9)\), \(K(-3,2)\)

\(JK=\sqrt{(-3 + 9)^2+(2 + 9)^2}=\sqrt{36+121}=\sqrt{157}\)

• For \(IK\): \(I(-2,-8)\), \(K(-3,2)\)

\(IK=\sqrt{(-3 + 2)^2+(2 + 8)^2}=\sqrt{1 + 100}=\sqrt{101}\)

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

• For \(RS\): \(R(8,8)\), \(S(1,1)\)

\(RS=\sqrt{(8 - 1)^2+(8 - 1)^2}=\sqrt{49+49}=\sqrt{98}\)

• For \(ST\): \(S(1,1)\), \(T(-1,8)\)

\(ST=\sqrt{(-1 - 1)^2+(8 - 1)^2}=\sqrt{4 + 49}=\sqrt{53}\)

• For \(RT\): \(R(8,8)\), \(T(-1,8)\)

\(RT=\sqrt{(-1 - 8)^2+(8 - 8)^2}=\sqrt{81+0}=9\)

Since the side - lengths of \(\triangle IJK\) (\(\sqrt{50},\sqrt{101},\sqrt{157}\)) and \(\triangle RST\) (\(\sqrt{98},\sqrt{53},9\)) are not equal.

Answer:

No, \(\triangle IJK\) and \(\triangle RST\) are not congruent.