Chapter 1.1, Problem 23E

a)

To determine

To find : the length of each sides of the given triangle.

Answer to Problem 23E

  $AB=13,BC=5andCA=12$

Explanation of Solution

Given information : we have the given triangle is

  

Let the vertices of the triangle be $A\left(1,0\right),B\left(13,5\right)andC\left(13,0\right)$

We know that the distance between two points $A\left(x_1,y_1\right)andB\left(x_2,y_2\right)$ is given by

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

Now, using the distance formula we get

  $\begin{array}{c}AB=\sqrt{{\left(1-13\right)}^2+{\left(0-5\right)}^2}\\ =\sqrt{{\left(-12\right)}^2+{\left(-5\right)}^2}\\ =\sqrt{144+25}\\ =\sqrt{169}\\ =13\end{array}$

  $\begin{array}{c}BC=\sqrt{{\left(13-13\right)}^2+{\left(5-0\right)}^2}\\ =\sqrt{{\left(0\right)}^2+{\left(5\right)}^2}\\ =\sqrt{25}\\ =5\end{array}$

  $\begin{array}{c}CA=\sqrt{{\left(13-1\right)}^2+{\left(0-0\right)}^2}\\ =\sqrt{{\left(12\right)}^2+{\left(0\right)}^2}\\ =\sqrt{144}\\ =12\end{array}$

b).

To determine

To Prove : that the triangle satisfy Pythagorean Theorem

Explanation of Solution

Given information : We have the given triangle is

  

For the vertices of the triangle be $A\left(1,0\right),B\left(13,5\right)andC\left(13,0\right)$

We get the sides of the triangle are $AB=13,BC=5andCA=12$

Considering the given triangle as a right angled triangle we see that

  $\begin{array}{l}A{B}^2=B{C}^2+C{A}^2\\ {\left(13\right)}^2={\left(5\right)}^2+{\left(12\right)}^2\end{array}$

Which proves the Pythagorean Theorem.