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 as a guide. let the three points (-6, -1), (-1, -3), and (5,2) be labeled as a, b, and c respectively. graph the triangle formed by the vertices a, b, and c. choose the correct graph below. a. graph b. graph c. graph d. graph 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 of the following must be true? 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.

Explanation:

To determine the correct option, we recall the property of perpendicular lines: two lines are perpendicular if and only if the product of their slopes is -1. In a right triangle, two sides are perpendicular, so their slopes must satisfy this condition.

• Option A: States that the product of the slopes of two sides is -1, which matches the perpendicular lines property.
• Option B: The sum of slopes being 0 means lines are parallel (since \(m_1 + m_2 = 0\) implies \(m_1=-m_2\)), not perpendicular.
• Option C: Same slope means lines are parallel, not perpendicular.
• Option D: Product of slopes being 1 does not indicate perpendicularity (perpendicularity requires product -1).

Thus, the correct option is A.

Answer:

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