Chapter 12, Problem 16CLT

To determine

To find: The lengths of the sides of the right triangle and to show that these lengths satisfy the Pythagorean Theorem.

Answer to Problem 16CLT

The lengths of the right triangle are 3,4,and 5

Explanation of Solution

Given triangle:

  

Calculation:

The vertices of given triangle are $A\left(0,0,0\right),B\left(0,0,3\right),C\left(0,4,3\right)$

We know that the distance between the points $\left(x_1,y_1,z_1\right)$ and $\left(x_2,y_2,z_2\right)$ is

  $d=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2+{\left(z_2-z_1\right)}^2}$

By using the distance formula , we get

  $\begin{array}{l}AB=\sqrt{{\left(0-0\right)}^2+{\left(0-0\right)}^2+{\left(3-0\right)}^2}=3\\ BC=\sqrt{{\left(0-0\right)}^2+{\left(4-0\right)}^2+{\left(3-3\right)}^2}=4\\ CA=\sqrt{{\left(0-0\right)}^2+{\left(4-0\right)}^2+{\left(3-0\right)}^2}=\sqrt{16+9}=5\end{array}$

Now,

  $\begin{array}{l}{\left(AB\right)}^2+{\left(BC\right)}^2={\left(3\right)}^2+{\left(4\right)}^2=25\\ {\left(AB\right)}^2+{\left(BC\right)}^2={\left(CA\right)}^2\end{array}$

Thus, the lengths of the triangle satisfy the Pythagorean Theorem.