Question

consider a right triangle, \\( \triangle abc \\), with vertices at \\( a(-4,3) \\), \\( b(2,1) \\), \\( c(3,4) \\). show that \\( \overline{ab} \perp \overline{bc} \\).
drag and drop the answers to the boxes to complete the sentences
the slope of \\( \overline{ab} \\) is
while the slope of \\( \overline{bc} \\) is
since the slopes of \\( \overline{ab} \\) and \\( \overline{bc} \\) are
we have shown that \\( \overline{ab} \perp \overline{bc} \\).

Explanation:

Step1: Calculate the slope of \(\overline{AB}\)

The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). For points \(A(-4,3)\) and \(B(2,1)\), \(m_{AB}=\frac{1 - 3}{2-(-4)}=\frac{-2}{6}=-\frac{1}{3}\)

Step2: Calculate the slope of \(\overline{BC}\)

For points \(B(2,1)\) and \(C(3,4)\), \(m_{BC}=\frac{4 - 1}{3 - 2}=\frac{3}{1}=3\)

Step3: Check the relationship between the slopes

The product of the slopes \(m_{AB}\times m_{BC}=-\frac{1}{3}\times3=- 1\). When the product of two non - zero slopes is \(-1\), the lines are perpendicular. The slopes \(m_{AB}=-\frac{1}{3}\) and \(m_{BC}=3\) are negative reciprocals of each other.

Answer:

The slope of \(\overline{AB}\) is \(-\frac{1}{3}\), while the slope of \(\overline{BC}\) is \(3\). Since the slopes of \(\overline{AB}\) and \(\overline{BC}\) are negative reciprocals of each other, we have shown that \(\overline{AB}\perp\overline{BC}\).