Question

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are \\( \triangle def \\) and \\( \triangle vwx \\) congruent?

Explanation:

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

Use the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) or count the units on the grid.
For \(DE\): Count the vertical units. \(DE= 3\) units.
For \(EF\): Count the horizontal units. \(EF = 8\) units.
For \(DF\): Using the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), \(DE = 3\), \(EF=8\), then \(DF=\sqrt{3^{2}+8^{2}}=\sqrt{9 + 64}=\sqrt{73}\)

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

For \(VW\): Count the horizontal units. \(VW = 8\) units.
For \(WX\): Count the vertical units. \(WX=9\) units.
For \(VX\): Using the Pythagorean theorem, \(VW = 8\), \(WX = 9\), then \(VX=\sqrt{8^{2}+9^{2}}=\sqrt{64+81}=\sqrt{145}\)

Since \(DE
eq WX\), \(EF = VW\) but \(DF
eq VX\), the triangles do not have all corresponding sides equal.

Answer:

No, \(\triangle DEF\) and \(\triangle VWX\) are not congruent.