Question

either prove or disprove the statement, \the points (-5, -3), (-2, -4), and (5,1) are the vertices of a right triangle.\ use a graph only as a guide. let the three points (-5, -3), (-2, -4), and (5,1) be labeled as a, b, and c respectively. graph the triangle formed by the vertices a, b, and c. choose the correct graph below. graphs a, b, c, d shown by looking at the graph it seems that the points (-5, -3), (-2, -4), and (5,1) do not form a right triangle. note that the right triangle has two sides perpendicular to each other. if (-5, -3), (-2, -4), and (5,1) are the vertices of a right triangle, which of the following must be true? a. two sides of the triangle must have the sum of their slopes as 0. b. two sides of the triangle must have the same slope. c. two sides of the triangle must have the product of their slopes as -1. d. two sides of the triangle must have the product of their slopes as 1.

Explanation:

To determine the correct option, we recall the property of perpendicular lines in terms of slopes. If two lines are perpendicular, the product of their slopes is -1. For a right triangle, two of its sides (which are lines) must be perpendicular, so their slopes should have a product of -1. Let's analyze each option:

• Option A: The sum of slopes being 0 is not a property of perpendicular lines. For example, slopes 1 and -1 sum to 0 but are perpendicular, but this is a coincidence. The key property is the product, not the sum. So A is incorrect.
• Option B: Having the same slope means the lines are parallel, not perpendicular. So B is incorrect.
• Option C: This matches the property of perpendicular lines (product of slopes is -1). So this is correct.
• Option D: A product of 1 for slopes means the lines are parallel (if slopes are equal) or have a reciprocal relationship but not perpendicular. Perpendicular lines have a product of -1, not 1. So D is incorrect.

Answer:

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