Question

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are \\( \triangle cde \\) and \\( \triangle wxy \\) congruent?

Explanation:

Step1: Calculate side lengths of \(\triangle CDE\)

Using the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
For \(CE\): \(C(4,-1)\), \(E(0,-8)\)
\(CE=\sqrt{(4 - 0)^2+(-1+8)^2}=\sqrt{16 + 49}=\sqrt{65}\)
For \(CD\): \(C(4,-1)\), \(D(8,-8)\)
\(CD=\sqrt{(8 - 4)^2+(-8 + 1)^2}=\sqrt{16+49}=\sqrt{65}\)
For \(ED\): \(E(0,-8)\), \(D(8,-8)\)
\(ED=\sqrt{(8 - 0)^2+(-8+8)^2}=8\)

Step2: Calculate side lengths of \(\triangle WXY\)

For \(WX\): \(W(6,6)\), \(X(0,1)\)
\(WX=\sqrt{(6 - 0)^2+(6 - 1)^2}=\sqrt{36+25}=\sqrt{61}\)
For \(WY\): \(W(6,6)\), \(Y(0,10)\)
\(WY=\sqrt{(6 - 0)^2+(6 - 10)^2}=\sqrt{36 + 16}=\sqrt{52}\)
For \(XY\): \(X(0,1)\), \(Y(0,10)\)
\(XY=\sqrt{(0 - 0)^2+(10 - 1)^2}=9\)

Since the side - lengths of \(\triangle CDE\) (\(\sqrt{65},\sqrt{65},8\)) and \(\triangle WXY\) (\(\sqrt{61},\sqrt{52},9\)) are not equal.

Answer:

No, \(\triangle CDE\) and \(\triangle WXY\) are not congruent.