Determine whether triangle DEF is congruent to triangle PQR given the coordinates of the vertices. Explain.

D(-6,1) E(1,2) F(-1,-4) P(0,5) Q(7,6) R(5,0)

To determine whether triangle DEF is congruent to triangle PQR, we need to compare their corresponding sides and angles.

1. Start by finding the lengths of the sides of triangle DEF and triangle PQR using the distance formula:

- Side DE:
- Distance = sqrt((xE - xD)^2 + (yE - yD)^2)
- Distance = sqrt((1 - (-6))^2 + (2 - 1)^2)
- Distance = sqrt(49 + 1)
- Distance = sqrt(50) = 5sqrt(2)

- Side DF:
- Distance = sqrt((xF - xD)^2 + (yF - yD)^2)
- Distance = sqrt((-1 - (-6))^2 + (-4 - 1)^2)
- Distance = sqrt(25 + 25)
- Distance = sqrt(50) = 5sqrt(2)

- Side EF:
- Distance = sqrt((xF - xE)^2 + (yF - yE)^2)
- Distance = sqrt((-1 - 1)^2 + (-4 - 2)^2)
- Distance = sqrt(4 + 36)
- Distance = sqrt(40) = 2sqrt(10)

- Side PQ:
- Distance = sqrt((xQ - xP)^2 + (yQ - yP)^2)
- Distance = sqrt((7 - 0)^2 + (6 - 5)^2)
- Distance = sqrt(49 + 1)
- Distance = sqrt(50) = 5sqrt(2)

- Side PR:
- Distance = sqrt((xR - xP)^2 + (yR - yP)^2)
- Distance = sqrt((5 - 0)^2 + (0 - 5)^2)
- Distance = sqrt(25 + 25)
- Distance = sqrt(50) = 5sqrt(2)

- Side QR:
- Distance = sqrt((xR - xQ)^2 + (yR - yQ)^2)
- Distance = sqrt((5 - 7)^2 + (0 - 6)^2)
- Distance = sqrt(4 + 36)
- Distance = sqrt(40) = 2sqrt(10)

2. Comparing the corresponding sides:
- Side DE is congruent to side PQ since they both have a length of 5sqrt(2).
- Side DF is congruent to side QR since they both have a length of 5sqrt(2).
- Side EF is congruent to side PR since they both have a length of 2sqrt(10).

3. Therefore, triangle DEF is congruent to triangle PQR because all three pairs of corresponding sides are congruent.

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