Question

plot each point and form the right triangle abc. verify that the triangle is a right triangle. find its area.
a=(2, - 5); b=(0, - 4); c=(6,3)
choose the correct graph below that shows points a, b, c, and triangle abc.
to show that the triangle is a right triangle, show that the sum of the squares of the lengths of two of the sides (the legs) equals the square of the length of the third side (the hypotenuse).
the sum of the squares of the lengths of the legs of the triangle is
the square of the length of the hypotenuse of the triangle is
find the area of the triangle.
the area of the triangle is square units.

Explanation:

Step1: Calculate the lengths of the sides

Use the distance formula $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.
For side $AB$ with $A=(2,-5)$ and $B=(0,-4)$:
$AB=\sqrt{(0 - 2)^2+(-4+5)^2}=\sqrt{(-2)^2+1^2}=\sqrt{4 + 1}=\sqrt{5}$
For side $BC$ with $B=(0,-4)$ and $C=(6,3)$:
$BC=\sqrt{(6 - 0)^2+(3 + 4)^2}=\sqrt{6^2+7^2}=\sqrt{36+49}=\sqrt{85}$
For side $AC$ with $A=(2,-5)$ and $C=(6,3)$:
$AC=\sqrt{(6 - 2)^2+(3 + 5)^2}=\sqrt{4^2+8^2}=\sqrt{16 + 64}=\sqrt{80}$

Step2: Verify the Pythagorean - theorem

$(AB)^2+(AC)^2=(\sqrt{5})^2+(\sqrt{80})^2=5 + 80=85$
$(BC)^2=(\sqrt{85})^2 = 85$
Since $(AB)^2+(AC)^2=(BC)^2$, the triangle is a right - triangle.

Step3: Calculate the area

The legs of the right - triangle are $AB$ and $AC$.
The area of a right - triangle $A=\frac{1}{2}\times\text{base}\times\text{height}$.
Here, base $=\sqrt{5}$ and height $=\sqrt{80}$, so $A=\frac{1}{2}\times\sqrt{5}\times\sqrt{80}=\frac{1}{2}\times\sqrt{5\times80}=\frac{1}{2}\times\sqrt{400}=\frac{1}{2}\times20 = 10$

Answer:

The sum of the squares of the lengths of the legs of the triangle is $85$.
The square of the length of the hypotenuse of the triangle is $85$.
The area of the triangle is $10$ square units.