Question

Plot each point and form the triangle ABC. Verify that the triangle is a right triangle. Find the area. A=(-5,3); B= ((6,0); C= (5,5) A=(4,-3); B=(4,1); C=(2,1)

Answer 1

1) It is a right triangle.

`Area=25.93\ units^2`

2) It is a right triangle.

`Area=4\ units^2`

The points of each triangle are plotted in the images attached.

Explanation:

1) The points
`A=(-5,3); B= ((6,0); C= (5,5)` are plotted in the first image attached.

Knowing the points of the triangle, you can find the slope of
`AC` and
`BC` with this formula:

`m=(y_2-y_1)/(x_2-x_1)`

Then:

`m_(AC)=(5-3)/(5-(-5))=(2)/(10)=(1)/(5)`

`m_(BC)=(5-0)/(5-6)=(5)/(-1)=-5`

Since the slopes of the sides
`AC` and
`BC` are negative reciprocals, they are perpendicular; therefore IT IS A RIGHT TRIANGLE.

Find the length of
`AC` and
`BC` in order to calculate the area of the triangle:

`AC=√((-5-5)^2+(3-5)^2)=10.19\ units\\\\BC=√((5-6)^2+(5-0)^2)=5.09\ units`

The area is:

`A=(AC*BC)/(2)=((10.19\ units)(5.09\ units))/(2)=25.93\ units^2`

2) The points
`A=(4,-3); B=(4,1); C=(2,1)` are plotted in the second image attached.

By definition horizontal and vertical lines are perpendicular, therefore IT IS A RIGHT TRIANGLE.

You can observe in the figure that the lenghts of the sides
`AB` and
`BC` are:

`AB=4\ units`

`BC=2\ units`

Therefore, the area is:

`A=(AB*BC)/(2)=((4\ units)(2\ units))/(2)=4\ units^2`