Question

step 1 of 2: determine whether or not the given points (2,5),(9, - 7) and (9,5) form a right triangle. if the triangle is not a right triangle, determine if it is isosceles or scalene.
answer
right triangle
isosceles triangle
scalene triangle

Explanation:

Step1: Calculate side - lengths using distance formula

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.
Let $A=(2,5)$, $B=(9, - 7)$ and $C=(9,5)$.
The length of $AB$ is $d_{AB}=\sqrt{(9 - 2)^2+(-7 - 5)^2}=\sqrt{7^2+( - 12)^2}=\sqrt{49 + 144}=\sqrt{193}$.
The length of $BC$ is $d_{BC}=\sqrt{(9 - 9)^2+(5+7)^2}=\sqrt{0 + 144}=12$.
The length of $AC$ is $d_{AC}=\sqrt{(9 - 2)^2+(5 - 5)^2}=\sqrt{7^2+0}=7$.

Step2: Check for right - triangle using Pythagorean theorem

The Pythagorean theorem states that for a right - triangle with side - lengths $a$, $b$ and hypotenuse $c$, $a^{2}+b^{2}=c^{2}$.
$7^{2}+12^{2}=49 + 144=193$, and $(\sqrt{193})^{2}=193$. So, $AC^{2}+BC^{2}=AB^{2}$.

Answer:

Right Triangle