Question

In Problems 101–104, find the length of each side of the triangle determined by the three points P1 , P2 , and P3 . State whether the triangle is an isosceles triangle, a right triangle, neither of these, or both. (An isosceles triangle is one in which at least two of the sides are of equal length.)

$P_1=\left(7,2\right),P_2=\left(-4,0\right),P_3=\left(4,6\right)$

Short answer

The length of sides of triangle are $5,10and5\sqrt{5}.$

The triangle is a Right Triangle.

Step-by-step solution

Step 1. Given Information

$P_1=\left(7,2\right),P_2=\left(-4,0\right),P_3=\left(4,6\right)$

Step 2. Distance Between P1 and P2

$d\left(P_1,P_2\right)=\sqrt{{\left(7+4\right)}^2+{\left(2-0\right)}^2}=\sqrt{125}=5\sqrt{5}$

Step 3. Distance between P2 and P3

$d\left(P_2,P_3\right)=\sqrt{{\left(-4-4\right)}^2+{\left(0-6\right)}^2}=\sqrt{100}=10$

Step 4. Distance between P1 and P3

$d\left(P_1,P_3\right)=\sqrt{{\left(7-4\right)}^2+{\left(2-6\right)}^2}=\sqrt{25}=5$

Step 5. Check for Right Triangle

$$\begin{gathered} 5^2+{10}^2={\left(5\sqrt{5}\right)}^2 \\ 25+100=125 \end{gathered}$$

The equation is true, the triangle is a right triangle.