Chapter 11.2, Problem 2E

To determine

To Find: The area and three heights of the triangle each correct to three significant digits.

Answer to Problem 2E

  $\begin{array}{l}A=17.321sq.unit\\ Heights:6.928,4.949,4.330\end{array}$

Explanation of Solution

Given: The length of the sides of a triangle are given.

  $5,7,8$

Given: $5,7,8$

i.e.

  $a=5,b=7,c=8$

Using Heron’s formula

  $\begin{array}{l}s=\frac{1}{2}(a+b+c)\\ s=\frac{1}{2}(5+7+8)\\ s=10\\ now\\ A=\sqrt{s(s-a)(s-b)(s-c)}\\ A=\sqrt{10(10-5)(10-7)(10-8)}\\ A=\sqrt{10(5)(3)(2)}\\ A=10\sqrt{3}\\ A=17.321\\ i.e.\\ h_1=\frac{2A}{b_1}=\frac{2A}{a}=\frac{2(17.321)}{5}=6.928\\ h_2=\frac{2A}{b_2}=\frac{2A}{b}=\frac{2(17.321)}{7}=4.949\\ h_3=\frac{2A}{b_3}=\frac{2A}{c}=\frac{2(17.321)}{8}=4.330\end{array}$

Hence,

  $\begin{array}{l}A=17.321sq.unit\\ Heights:6.928,4.949,4.330\end{array}$