Question

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are \\( \triangle b c d \\) and \\( \triangle i j k \\) congruent?

Explanation:

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

Use the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
For \(BC\): \(B(10,9)\), \(C(0,9)\), \(BC=\sqrt{(10 - 0)^2+(9 - 9)^2}=10\).
For \(CD\): \(C(0,9)\), \(D(10,0)\), \(CD=\sqrt{(10 - 0)^2+(0 - 9)^2}=\sqrt{100 + 81}=\sqrt{181}\).
For \(BD\): \(B(10,9)\), \(D(10,0)\), \(BD=\sqrt{(10 - 10)^2+(0 - 9)^2}=9\).

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

For \(IJ\): \(I(0,- 9)\), \(J(0,0)\), \(IJ=\sqrt{(0 - 0)^2+(0+9)^2}=9\).
For \(JK\): \(J(0,0)\), \(K(-10,0)\), \(JK=\sqrt{(-10 - 0)^2+(0 - 0)^2}=10\).
For \(IK\): \(I(0,-9)\), \(K(-10,0)\), \(IK=\sqrt{(-10 - 0)^2+(0 + 9)^2}=\sqrt{100+81}=\sqrt{181}\).

Step3: Compare the side lengths

Since \(BC = JK = 10\), \(CD=IK=\sqrt{181}\), \(BD = IJ=9\), by SSS (Side - Side - Side) congruence criterion.

Answer:

\(\triangle BCD\) and \(\triangle IJK\) are congruent.