Question

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are \\( \triangle hij \\) and \\( \triangle xyz \\) congruent?

Explanation:

Step1: Find the length of sides using distance formula

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

• \(XY\): \(X(- 3,6)\), \(Y(-10,0)\)

\(XY=\sqrt{(-3+10)^2+(6 - 0)^2}=\sqrt{49 + 36}=\sqrt{85}\)

• \(XZ\): \(X(-3,6)\), \(Z(-1,0)\)

\(XZ=\sqrt{(-3 + 1)^2+(6-0)^2}=\sqrt{4+36}=\sqrt{40}\)

• \(YZ\): \(Y(-10,0)\), \(Z(-1,0)\)

\(YZ=\sqrt{(-1 + 10)^2+(0-0)^2}=9\)

For \(\triangle HIJ\):

• \(HI\): \(H(1,9)\), \(I(8,1)\)

\(HI=\sqrt{(8 - 1)^2+(1 - 9)^2}=\sqrt{49+64}=\sqrt{113}\)

• \(HJ\): \(H(1,9)\), \(J(8,10)\)

\(HJ=\sqrt{(8 - 1)^2+(10 - 9)^2}=\sqrt{49 + 1}=\sqrt{50}\)

• \(IJ\): \(I(8,1)\), \(J(8,10)\)

\(IJ=\sqrt{(8 - 8)^2+(10 - 1)^2}=9\)

Step2: Compare the side - lengths

Since \(\sqrt{85}
eq\sqrt{113}\), \(\sqrt{40}
eq\sqrt{50}\), although \(YZ = IJ=9\), the other corresponding sides are not equal.

Answer:

\(\triangle HIJ\) and \(\triangle XYZ\) are not congruent.