Question

the graph shows triangles cde and wxy. is cde congruent to wxy? justify your answer.

Explanation:

Step1: Find the lengths of the sides of triangle \(WXY\)

Use the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
For \(W(-4,6)\), \(X(-7,7)\), \(Y(-8,2)\)

• Length of \(WX\): \(\sqrt{(-7+4)^2+(7 - 6)^2}=\sqrt{(-3)^2+1^2}=\sqrt{9 + 1}=\sqrt{10}\)
• Length of \(XY\): \(\sqrt{(-8 + 7)^2+(2-7)^2}=\sqrt{(-1)^2+(-5)^2}=\sqrt{1+25}=\sqrt{26}\)
• Length of \(WY\): \(\sqrt{(-8 + 4)^2+(2-6)^2}=\sqrt{(-4)^2+(-4)^2}=\sqrt{16+16}=\sqrt{32}=4\sqrt{2}\)

Step2: Find the lengths of the sides of triangle \(CDE\)

For \(C(8,7)\), \(D(6,9)\), \(E(5,4)\)

• Length of \(CD\): \(\sqrt{(6 - 8)^2+(9 - 7)^2}=\sqrt{(-2)^2+2^2}=\sqrt{4+4}=\sqrt{8}=2\sqrt{2}\)
• Length of \(DE\): \(\sqrt{(5 - 6)^2+(4 - 9)^2}=\sqrt{(-1)^2+(-5)^2}=\sqrt{1 + 25}=\sqrt{26}\)
• Length of \(CE\): \(\sqrt{(5 - 8)^2+(4 - 7)^2}=\sqrt{(-3)^2+(-3)^2}=\sqrt{9+9}=\sqrt{18}=3\sqrt{2}\)

Step3: Compare the side lengths

Since the side lengths of \(\triangle WXY\) (\(\sqrt{10},\sqrt{26},4\sqrt{2}\)) and \(\triangle CDE\) (\(2\sqrt{2},\sqrt{26},3\sqrt{2}\)) are not equal.

Answer:

No, \(\triangle CDE\) is not congruent to \(\triangle WXY\) because their corresponding side lengths are not equal.