DESCRIBE IN WORDS what you would calculate and how you would determine whether the triangle with vertices A (2,1), B (5,3), and C (0,4

Question

GPT 3.5 · Answer

To determine whether the triangle formed by the vertices A (2,1), B (5,3), and C (0,4) is a right triangle or not, you would calculate the slopes of two of its sides and check if they are negative reciprocals of each other. Additionally, you could also calculate the lengths of the sides and verify if they satisfy the Pythagorean theorem.

Here are the steps to determine if the triangle is a right triangle:

1. Calculate the slopes of the two sides formed by the given vertices. The slope of a line passing through two points (x1, y1) and (x2, y2) is given by the formula: slope = (y2 - y1) / (x2 - x1).

- Slope of side AB:
slope_AB = (3 - 1) / (5 - 2) = 2 / 3

- Slope of side BC:
slope_BC = (4 - 3) / (0 - 5) = 1 / (-5) = -1/5

2. Check if the slopes are negative reciprocals of each other. If they are, then the triangle is a right triangle. In this case, the slopes are not negative reciprocals of each other (2/3 ≠ -1/5), so we move to the next step.

3. Calculate the lengths of the three sides using the distance formula. The distance between two points (x1, y1) and (x2, y2) is given by the formula: distance = √((x2 - x1)^2 + (y2 - y1)^2).

- Length of side AB:
length_AB = √((5 - 2)^2 + (3 - 1)^2) = √(9 + 4) = √13

- Length of side BC:
length_BC = √((0 - 5)^2 + (4 - 3)^2) = √(25 + 1) = √26

- Length of side AC:
length_AC = √((0 - 2)^2 + (4 - 1)^2) = √(4 + 9) = √13

4. Check if the lengths of the sides satisfy the Pythagorean theorem. If the sum of the squares of the two shorter sides is equal to the square of the longest side, then the triangle is a right triangle. In this case, √13^2 + √26^2 = √13^2, which is not true. Hence, the triangle formed by the given vertices A, B, and C is not a right triangle.

In conclusion, by calculating the slopes and lengths of the sides, and checking the Pythagorean theorem, we can determine that the triangle with vertices A (2,1), B (5,3), and C (0,4) is not a right triangle.