Question

either prove or disprove the statement, \the points (-6, -1), (-1, -3), and (5,2) are the vertices of a right triangle.\ use a graph only a by looking at the graph it seems that the points (-6, -1), (-1, -3), and (5,2) do not form a right triangle. note that the right triangle has two sides perpendicular to each other. if (-6, -1), (-1, -3), and (5,2) are the vertices of a right triangle, which o a. two sides of the triangle must have the product of their slopes as -1. b. two sides of the triangle must have the sum of their slopes as 0. c. two sides of the triangle must have the same slope. d. two sides of the triangle must have the product of their slopes as 1. what is the formula for the slope of a line through \\((x_1,y_1)\\) and \\((x_2,y_2)\\) with \\(x_1 \
eq x_2\\)? a. \\(\frac{y_1 - x_1}{y_2 - x_2}\\) b. \\(\frac{x_2 - x_1}{y_2 - y_1}\\) c. \\(\frac{y_2 - y_1}{x_2 - x_1}\\) d. \\(\frac{y_2 - x_2}{y_1 - x_1}\\)

Explanation:

First Question (Multiple Choice about Right Triangle Slopes)

To determine if two lines are perpendicular, their slopes' product is -1 (since perpendicular lines have slopes that are negative reciprocals). So for a right triangle, two sides (lines) must be perpendicular, meaning the product of their slopes is -1. Option A states this, while B (sum 0 is for opposite slopes, not perpendicular), C (same slope is parallel), D (product 1 is not perpendicular) are incorrect.

The slope \( m \) of a line through points \( (x_1, y_1) \) and \( (x_2, y_2) \) (with \( x_1
eq x_2 \)) is calculated as the change in \( y \) over the change in \( x \), which is \( \frac{y_2 - y_1}{x_2 - x_1} \). Option C matches this formula. Option A, B, D have incorrect numerator/denominator combinations.

Answer:

A. Two sides of the triangle must have the product of their slopes as -1.

Second Question (Slope Formula)