Question

recall that the three points $(-5, -3)$, $(-2, -4)$, and $(5,1)$ are labeled as $a$, $b$, and $c$ respectively. find the slopes of each side of the triangle. slope $m_{ab}$ of the side $ab = -\frac{1}{3}$ (type an integer or a simplified fraction.) slope $m_{bc}$ of the side $bc = \frac{5}{7}$ (type an integer or a simplified fraction.) slope $m_{ac}$ of the side $ac = \frac{2}{5}$ (type an integer or a simplified fraction.) find $m_{ab} cdot m_{bc}$, $m_{ab} cdot m_{ac}$, and $m_{bc} cdot m_{ac}$. $m_{ab} cdot m_{bc} = -\frac{5}{21}$ $m_{ab} cdot m_{ac} = -\frac{2}{15}$ $m_{bc} cdot m_{ac} = \frac{2}{7}$ (type integers or simplified fractions.) does the triangle have two perpendicular sides? $\bigcirc$ yes $checkmark$ no is the statement \the points $(-5, -3)$, $(-2, -4)$, and $(5,1)$ are the vertices of a right triangle.\ proved or disproved? $\bigcirc$ a. the statement is disproved because no two sides of the triangle form a right angle. $\bigcirc$ b. the statement is disproved because two sides are parallel. $\bigcirc$ c. the statement is proved because two sides of the triangle form a right angle. $\bigcirc$ d. the statement is proved because no two sides are parallel.

Explanation:

Step1: Recall perpendicular slope condition

Two lines are perpendicular if the product of their slopes is \(-1\). We have the products: \(m_{AB}\cdot m_{BC}=-\frac{5}{21}\), \(m_{AB}\cdot m_{AC}=-\frac{2}{15}\), \(m_{BC}\cdot m_{AC}=\frac{2}{7}\). None of these products equal \(-1\), so no two sides are perpendicular.

Step2: Analyze the statement

A right triangle has two perpendicular sides (forming a right angle). Since no two sides here are perpendicular, the statement "The points \((-5, -3)\), \((-2, -4)\), and \((5,1)\) are the vertices of a right triangle" is disproved because no two sides form a right angle.

Answer:

A. The statement is disproved because no two sides of the triangle form a right angle.