Question

Determine whether △DEF≅△PQR given the coordinates of the vertices. A. D(–6, 1), E(1, 2), F(–1, –4), P(0, 5), Q(7, 6), R(5, 0)

Answer 1

Using the distance formula to calculate distance between vertices, our calculation shows that △DEF and △PQR both have corresponding side lengths that are equal in size.

• Therefore,  △DEF ≅ △PQR.

Recall:

• Distance between two vertices given their coordinates is calculated using the distance formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

• Two congruent triangles will have corresponding sides that have equal lengths.

Find the side lengths of △DEF:

Distance between D(–6, 1) and E(1, 2) using $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

• Substitute

$DE = \sqrt{(1 -(-6))^2 + (2 - 1)^2} \\\\DE = \sqrt{7^2 + 1^2} \\\\DE = \sqrt{50}$

Distance between D(–6, 1) and F(–1, –4):

• Substitute

$DF = \sqrt{(-1 -(-6))^2 + (-4 - 1)^2} \\\\DF = \sqrt{5^2 + (-5^2)} \\\\DF = \sqrt{50}$

Distance between E(1, 2) and F(–1, –4):

• Applying the same formula and step above, $\mathbf{EF = \sqrt{40} }$

Find the side lengths of △PQR:

Also, using the distance formula, the following are the side lengths of △PQR,

Distance between P(0, 5) and Q(7, 6): $\mathbf{PQ = \sqrt{50}}$

Distance between P(0, 5) and R(5, 0): $\mathbf{PR = \sqrt{50}}$

Distance between Q(7, 6) and R(5, 0): $\mathbf{PR = \sqrt{40}}$

From our calculation, it shows that △DEF and △PQR both have corresponding side lengths that are equal in size.

• Therefore,  △DEF ≅ △PQR.

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