Q. 101 - Precalculus Enhanced with Graphing Utilities

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.)

$P_1=\left(2,1\right),P_2=\left(-4,1\right),P_3=\left(-4,-3\right)$

Step-by-Step Solution

Answer

The length of each side of triangle is $6,4 and 2\sqrt{13}.$

The triangle is a Right triangle.

Step 1. Given Information

$P_1=\left(2,1\right),P_2=\left(-4,1\right),P_3=\left(-4,-3\right)$

Step 2. Distance between P 1 and P 2

$d\left(P_1,P_2\right)=\sqrt{{\left(2+4\right)}^2+{\left(1-1\right)}^2}=\sqrt{36}=6$

Step 3. Distance between P 2   a n d   P 3

$d\left(P_2,P_3\right)=\sqrt{{\left(-4+4\right)}^2+{\left(1+3\right)}^2}=\sqrt{16}=4$

Step 4. Distance between P 1   a n d   P 3 .

$d\left(P_1,P_3\right)=\sqrt{{\left(2+4\right)}^2+{\left(1+3\right)}^2}=\sqrt{36+16}=2\sqrt{13}$

Step 5. Checking to see Right Triangle

$6^2+4^2={\left(2\sqrt{13}\right)}^2$

$$\begin{gathered} 36+16=52 \\ 52=52 \end{gathered}$$