Question

plot each point and form the triangle abc. verify that the triangle abc is a right triangle. find its area. a = (-4,7) b = (1,5) c = (-1,0) choose the correct graph that shows points a, b, c, and triangle abc.

Explanation:

Step1: Calculate the slopes of the sides

The slope formula is $m = \frac{y_2 - y_1}{x_2 - x_1}$.
For side $AB$ with $A(-4,7)$ and $B(1,5)$:
$m_{AB}=\frac{5 - 7}{1-(-4)}=\frac{-2}{5}=-\frac{2}{5}$
For side $BC$ with $B(1,5)$ and $C(-1,0)$:
$m_{BC}=\frac{0 - 5}{-1 - 1}=\frac{-5}{-2}=\frac{5}{2}$
For side $AC$ with $A(-4,7)$ and $C(-1,0)$:
$m_{AC}=\frac{0 - 7}{-1-(-4)}=\frac{-7}{3}=-\frac{7}{3}$
Since $m_{AB}\times m_{BC}=-\frac{2}{5}\times\frac{5}{2}=- 1$, sides $AB$ and $BC$ are perpendicular, so $\triangle ABC$ is a right - triangle.

Step2: Calculate the lengths of the two perpendicular sides

The distance formula is $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.
For $AB$:
$d_{AB}=\sqrt{(1 + 4)^2+(5 - 7)^2}=\sqrt{25 + 4}=\sqrt{29}$
For $BC$:
$d_{BC}=\sqrt{(-1 - 1)^2+(0 - 5)^2}=\sqrt{4 + 25}=\sqrt{29}$

Step3: Calculate the area of the right - triangle

The area formula for a right - triangle is $A=\frac{1}{2}\times base\times height$.
$A=\frac{1}{2}\times\sqrt{29}\times\sqrt{29}=\frac{29}{2}=14.5$

Answer:

The triangle $ABC$ is a right - triangle and its area is $14.5$. (Note: To choose the correct graph, plot the points $A(-4,7)$, $B(1,5)$ and $C(-1,0)$ on the coordinate plane. The correct graph will have these three points connected to form a right - triangle. Without actually plotting on a graphing tool, we can't definitively pick from A, B, C, D here based on the text - only the verification of right - triangle and area calculation can be fully completed in this format)