Answer:To verify whether triangle DEF is a right triangle, we can use the slope formula and the perpendicularity criterion.
a) Verification of DEF as a right triangle:
Calculate the slopes of the two sides:
Slope of DE = (y2 - y1) / (x2 - x1) = (4 - 2) / (1 - (-2)) = 2 / 3
Slope of EF = (y2 - y1) / (x2 - x1) = (1 - 2) / (3 - (-2)) = -1 / 5
Check if the product of the slopes is -1 (perpendicular lines):
(Slope of DE) * (Slope of EF) = (2 / 3) * (-1 / 5) = -2 / 15
Since the product of the slopes is not -1, the sides DE and EF are not perpendicular. Therefore, triangle DEF is not a right triangle.
b) Another method to determine if triangle DEF is a right triangle:
Another method to answer part a) is by calculating the lengths of the three sides DE, EF, and DF. Then, we can check if the Pythagorean theorem holds true.
Calculate the lengths of the sides:
Length of DE = sqrt((x2 - x1)^2 + (y2 - y1)^2) = sqrt((1 - (-2))^2 + (4 - 2)^2) = sqrt(9 + 4) = sqrt(13)
Length of EF = sqrt((x2 - x1)^2 + (y2 - y1)^2) = sqrt((3 - 1)^2 + (1 - 4)^2) = sqrt(4 + 9) = sqrt(13)
Length of DF = sqrt((x2 - x1)^2 + (y2 - y1)^2) = sqrt((3 - (-2))^2 + (1 - 2)^2) = sqrt(25 + 1) = sqrt(26)
Apply the Pythagorean theorem:
If DF^2 = DE^2 + EF^2, then triangle DEF is a right triangle.
DF^2 = (sqrt(26))^2 = 26
DE^2 + EF^2 = (sqrt(13))^2 + (sqrt(13))^2 = 13 + 13 = 26
Since DF^2 equals DE^2 + EF^2, the Pythagorean theorem holds true, indicating that triangle DEF is a right triangle.
Therefore, the two methods of verification provide conflicting results. One method (using the perpendicularity criterion) suggests that DEF is not a right triangle, while the other method (using the Pythagorean theorem) suggests that it is a right triangle. In such cases, it is important to double-check the calculations and verify the accuracy of the given points to resolve the discrepancy.
Explanation: