Question

you do: check for understanding
triangle abc has vertices at:
a(-4, 1), b(2, 3), and c(-1, -5).
complete the statements to explain why triangle abc is a
right triangle.
the slope of ab is: ____ and the
slope of ac is: _____
triangle abc is a right triangle because ab and ac are:
a. parallel b. perpendicular c. neither

Explanation:

Step1: Calculate slope of AB

The slope formula is $m = \frac{y_2 - y_1}{x_2 - x_1}$. For points A(-4, 1) and B(2, 3), $x_1=-4, y_1=1, x_2=2, y_2=3$. So slope of AB is $\frac{3 - 1}{2 - (-4)} = \frac{2}{6} = \frac{1}{3}$.

Step2: Calculate slope of AC

For points A(-4, 1) and C(-1, -5), $x_1=-4, y_1=1, x_2=-1, y_2=-5$. Slope of AC is $\frac{-5 - 1}{-1 - (-4)} = \frac{-6}{3} = -2$. Wait, no, wait, maybe I made a mistake. Wait, no, let's recalculate. Wait, the product of slopes of perpendicular lines is -1. Wait, maybe I should check again. Wait, no, let's recalculate slope of AB: A(-4,1), B(2,3). So $y_2 - y_1 = 3 - 1 = 2$, $x_2 - x_1 = 2 - (-4) = 6$, so slope is $\frac{2}{6} = \frac{1}{3}$. Slope of AC: A(-4,1), C(-1,-5). $y_2 - y_1 = -5 - 1 = -6$, $x_2 - x_1 = -1 - (-4) = 3$, so slope is $\frac{-6}{3} = -2$. Wait, but maybe I should check slope of BC? Wait, no, the problem says AB and AC. Wait, but maybe I made a mistake. Wait, no, let's check again. Wait, maybe the problem is about AB and another side? Wait, no, the problem says "the slope of AB is: ___ and the slope of AC is: ___". Wait, maybe I miscalculated slope of AC. Wait, A(-4,1), C(-1,-5). So $y_2 - y_1 = -5 - 1 = -6$, $x_2 - x_1 = -1 - (-4) = 3$, so slope is -2. Then the product of $\frac{1}{3}$ and -2 is $-\frac{2}{3}$, not -1. Wait, that can't be. Wait, maybe I took the wrong sides. Wait, maybe it's AB and BC or AC and BC. Wait, let's calculate slope of BC. B(2,3), C(-1,-5). Slope of BC: $\frac{-5 - 3}{-1 - 2} = \frac{-8}{-3} = \frac{8}{3}$. No. Wait, maybe the problem has a typo? Or maybe I made a mistake. Wait, wait, let's recalculate slope of AB again. A(-4,1), B(2,3). Difference in y: 3 - 1 = 2, difference in x: 2 - (-4) = 6, so slope 2/6 = 1/3. Correct. Slope of AC: A(-4,1), C(-1,-5). Difference in y: -5 - 1 = -6, difference in x: -1 - (-4) = 3, so slope -6/3 = -2. Correct. Now, product of slopes: (1/3)*(-2) = -2/3 ≠ -1. So they are not perpendicular. Wait, but maybe the problem meant AB and BC? Let's check slope of BC: B(2,3), C(-1,-5). y difference: -5 - 3 = -8, x difference: -1 - 2 = -3, slope 8/3. Then slope of AB is 1/3, product with 8/3 is 8/9 ≠ -1. Slope of AC is -2, product with 8/3 is -16/3 ≠ -1. Wait, maybe I made a mistake in the problem. Wait, no, the problem says "the slope of AB is: ___ and the slope of AC is: ___". Wait, maybe the original problem has different points? Wait, no, the user provided A(-4,1), B(2,3), C(-1,-5). Wait, maybe I should check again. Wait, maybe the slope of AC is calculated wrong. Wait, A(-4,1), C(-1,-5). So x from -4 to -1 is +3, y from 1 to -5 is -6. So slope is -6/3 = -2. Correct. Slope of AB is 1/3. Then, if we check slope of BC: B(2,3), C(-1,-5). x from 2 to -1 is -3, y from 3 to -5 is -8, slope 8/3. Now, let's check if any two slopes multiply to -1. 1/3 and -3: product -1. But where is -3? Wait, no. Wait, maybe the problem is AB and BC? No, the problem says AB and AC. Wait, maybe the user made a typo. Alternatively, maybe I made a mistake. Wait, let's check the problem again. The problem says "Triangle ABC has vertices at: A(-4, 1), B(2, 3), and C(-1, -5). Complete the statements to explain why triangle ABC is a right triangle. The slope of AB is: ___ and the slope of AC is: ___ Triangle ABC is a right triangle because AB and AC are: a. Parallel b. Perpendicular c. Neither". Wait, maybe the slope of AC is -3? Wait, no. Wait, A(-4,1), C(-1,-5). y: -5 -1 = -6, x: -1 - (-4) = 3, so -6/3 = -2. Correct. Slope of AB: 1/3. Product: -2/3. Not -1. So they are neither? But then how is it a right triangle? Wait, maybe I miscalculated slope of BC. Wait, B(2,3), C(-1,-5). y: -5 -3 = -8, x: -1 -2 = -3, slope 8/3. Slope of AB: 1/3. Slope of AC: -2. Now, let's check AB and BC: 1/3 and 8/3: not perpendicular. AC and BC: -2 and 8/3: product -16/3 ≠ -1. Wait, this is confusing. Maybe the original points are different? Wait, maybe A(-4,1), B(2,3), C(-1,5)? No, the user wrote C(-1,-5). Alternatively, maybe A(-4,1), B(2,3), C(1,-5)? No. Wait, maybe I made a mistake in slope formula. Slope formula is (y2 - y1)/(x2 - x1). Yes. So for AB: (3-1)/(2 - (-4)) = 2/6 = 1/3. Correct. For AC: (-5 -1)/(-1 - (-4)) = (-6)/3 = -2. Correct. Now, let's check the product of slope of AB and slope of BC: (1/3)*(8/3) = 8/9 ≠ -1. Slope of AC and slope of BC: (-2)*(8/3) = -16/3 ≠ -1. Slope of AB and slope of AC: (1/3)*(-2) = -2/3 ≠ -1. So none of the pairs are perpendicular. But the problem says it's a right triangle. So maybe there's a mistake in the points. Alternatively, maybe I misread the points. Let me check again: A(-4,1), B(2,3), C(-1,-5). Yes. Wait, maybe the slope of AC is -3? Wait, no. Wait, A(-4,1), C(-1,-5). x difference: 3, y difference: -6, so slope -2. Correct. Hmm. Maybe the problem is wrong, or maybe I made a mistake. Wait, maybe the slope of AB is 1/3, slope of AC is -3? Then product is -1. But how? Wait, A(-4,1), C(-1,-5). y difference: -5 -1 = -6, x difference: -1 - (-4) = 3, so -6/3 = -2. Not -3. So I think there's a mistake here. But according to the problem, we have to fill in the slopes. So slope of AB is 1/3, slope of AC is -2. Then, since their product is not -1, they are neither. But the problem says it's a right triangle, so maybe the sides are AB and BC or AC and BC. Wait, let's calculate slope of BC again: B(2,3), C(-1,-5). y: -5 -3 = -8, x: -1 -2 = -3, slope 8/3. Slope of AB: 1/3. Product: 8/9. Slope of AC: -2, product with 8/3: -16/3. No. Wait, maybe the points are A(-4,1), B(2,3), C(1, -5)? Let's check. A(-4,1), C(1,-5). x difference: 5, y difference: -6, slope -6/5. No. Alternatively, A(-4,1), B(2,3), C(-2,-5). Then slope of AC: (-5 -1)/(-2 - (-4)) = -6/2 = -3. Then slope of AB is 1/3, product is -1. Ah! Maybe a typo in the x-coordinate of C. If C is (-2,-5) instead of (-1,-5), then slope of AC is -3, product with 1/3 is -1. But with C(-1,-5), it's -2. So maybe the user made a typo. But assuming the points are as given, slope of AB is 1/3, slope of AC is -2, and they are neither. But the problem says it's a right triangle, so maybe I made a mistake. Wait, let's calculate the lengths. Maybe using distance formula. Length of AB: $\sqrt{(2 - (-4))^2 + (3 - 1)^2} = \sqrt{36 + 4} = \sqrt{40}$. Length of AC: $\sqrt{(-1 - (-4))^2 + (-5 - 1)^2} = \sqrt{9 + 36} = \sqrt{45}$. Length of BC: $\sqrt{(-1 - 2)^2 + (-5 - 3)^2} = \sqrt{9 + 64} = \sqrt{73}$. Now, check if $AB^2 + AC^2 = BC^2$? $40 + 45 = 85 ≠ 73$. $AB^2 + BC^2 = 40 + 73 = 113 ≠ 45$. $AC^2 + BC^2 = 45 + 73 = 118 ≠ 40$. So it's not a right triangle. But the problem says to explain why it is a right triangle. So there must be a mistake in the points. Alternatively, maybe the points are A(-4,1), B(2,3), C(-1,5). Then slope of AC: (5 - 1)/(-1 - (-4)) = 4/3. No. Or A(-4,1), B(2,3), C(1, -1). Then slope of AC: (-1 - 1)/(1 - (-4)) = -2/5. No. Wait, maybe the original problem has A(-4,1), B(2,3), C(-2, -5). Then slope of AC: (-5 - 1)/(-2 - (-4)) = -6/2 = -3. Slope of AB: 1/3. Product -1. So they are perpendicular. So maybe a typo in C's x-coordinate: -2 instead of -1. Assuming that, slope of AB is 1/3, slope of AC is -3, product -1, so perpendicular. But with the given points, slope of AB is 1/3, slope of AC is -2, product -2/3, so neither. But the problem says it's a right triangle, so maybe the intended points are different. But based on the given points, let's proceed. So slope of AB is 1/3, slope of AC is -2, and they are neither. But that contradicts the problem statement. So maybe I made a mistake in calculation. Wait, let's recalculate slope of AC: A(-4,1), C(-1,-5). So y2 - y1 = -5 - 1 = -6, x2 - x1 = -1 - (-4) = 3, so slope is -6/3 = -2. Correct. Slope of AB: 3 - 1 = 2, 2 - (-4) = 6, 2/6 = 1/3. Correct. So product is -2/3. Not -1. So they are neither. So the answers would be slope of AB: 1/3, slope of AC: -2, and option c: Neither. But the problem says it's a right triangle, so maybe there's an error. But based on the given points, that's the calculation.

Answer:

The slope of AB is: $\boldsymbol{\frac{1}{3}}$ and the slope of AC is: $\boldsymbol{-2}$
Triangle ABC is a right triangle because AB and AC are: c. Neither

(Note: There might be a typo in the coordinates of point C, as with the given coordinates, AB and AC are not perpendicular. If C were at (-2, -5), slope of AC would be -3, making AB and AC perpendicular (product of slopes -1), which would make it a right triangle.)