Question

1. the diagram shows triangle abc. which of the following statements would prove that triangle abc is a right triangle? a. (slope ab) times (slope bc)=1 b. (slope ab) times (slope bc)= -1 c. distance from a to b = distance from b to c d. distance from a to b=-(distance from b to c)

Explanation:

Step1: Recall perpendicular - slope property

In a right - triangle, if two sides are perpendicular, the product of their slopes is - 1.

Step2: Analyze each option

For option A, if (Slope \(AB\))\(\times\)(Slope \(BC\)) = 1, the lines are not perpendicular. For option B, if (Slope \(AB\))\(\times\)(Slope \(BC\))=-1, then \(AB\perp BC\), which means \(\angle B = 90^{\circ}\) and \(\triangle ABC\) is a right - triangle. For option C, distance from \(A\) to \(B\) = distance from \(B\) to \(C\) only shows that two sides are of equal length (isosceles property), not a right - triangle property. For option D, distance is non - negative, so distance from \(A\) to \(B=-( \text{distance from }B\text{ to }C)\) is not a valid statement.

Answer:

B. (Slope \(AB\)) times (slope \(BC\)) = - 1