Chapter 11.1, Problem 39E

To determine

Find the lengths of the sides of the right triangle with the indicated vertices.

Answer to Problem 39E

  $\boxed{45=45}$

Explanation of Solution

Given information:

Find the lengths of the sides of the right triangle with the indicated vertices. Show that these lengths satisfy the Pythagorean Theorem.

  $\left(0,0,0,\right),\left(2,2,1\right),\left(2,-4,4\right)$

Calculation:

Consider the given points.

  $\left(0,0,0,\right),\left(2,2,1\right),\left(2,-4,4\right)$

Consider the right angle triangle,

  

Find distance AB,BC,CA by using distance formula,

  $\begin{array}{l}AB=\sqrt{{\left(2-0\right)}^2+{\left(2-0\right)}^2+{\left(1-0\right)}^2}\\ =\sqrt{4+4+1}\\ =\sqrt{9}\\ =3\end{array}$

  $\begin{array}{l}BC=\sqrt{{\left(2-2\right)}^2+{\left(-4-2\right)}^2+{\left(4-1\right)}^2}\\ =\sqrt{0+36+9}\\ =\sqrt{45}\\ =3\sqrt{5}\end{array}$

  $\begin{array}{l}CA=\sqrt{{\left(0-2\right)}^2+{\left(0-4\right)}^2+{\left(0-4\right)}^2}\\ =\sqrt{4+16+16}\\ =\sqrt{36}\end{array}$

Now according to Pythagorean Theorem,

  $\begin{array}{l}C{A}^2+A{B}^2=B{C}^2\\ 6^2+3^2={\left(3\sqrt{5}\right)}^2\\ 45={\left(\sqrt{45}\right)}^2\\ 45=45\end{array}$

Hence, it satisfy Pythagorean Theorem with right angle at A $\boxed{45=45}$