Question

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are \\( \triangle efg \\) and \\( \triangle bcd \\) congruent?

Explanation:

Step1: Calculate the side lengths using the distance formula

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

• \(EF\): \(E(-10,9)\), \(F(-10,0)\), \(EF=\sqrt{(-10 + 10)^2+(9 - 0)^2}=9\)
• \(FG\): \(F(-10,0)\), \(G(2,9)\), \(FG=\sqrt{(2 + 10)^2+(9 - 0)^2}=\sqrt{144 + 81}=\sqrt{225}=15\)
• \(EG\): \(E(-10,9)\), \(G(2,9)\), \(EG = 2+10 = 12\)

For \(\triangle BCD\):

• \(BC\): \(B(1,-10)\), \(C(10,-10)\), \(BC=10 - 1=9\)
• \(CD\): \(C(10,-10)\), \(D(1,2)\), \(CD=\sqrt{(1 - 10)^2+(2 + 10)^2}=\sqrt{81+144}=\sqrt{225}=15\)
• \(BD\): \(B(1,-10)\), \(D(1,2)\), \(BD=2 + 10=12\)

Step2: Check the congruence

Since \(EF = BC = 9\), \(FG = CD = 15\), \(EG = BD = 12\), by SSS (Side - Side - Side) congruence criterion.

Answer:

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