Chapter 11.2, Problem 1E

To determine

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

Answer to Problem 1E

  $\begin{array}{l}A=42.426sq.unit\\ Heights:9.428,8.485,7.713\end{array}$

Explanation of Solution

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

  $9,10,11$

Given: $9,10,11$

i.e.

  $a=9,b=10,c=11$

Using Heron’s formula

  $\begin{array}{l}s=\frac{1}{2}(a+b+c)\\ s=\frac{1}{2}(9+10+11)\\ s=15\\ now\\ A=\sqrt{s(s-a)(s-b)(s-c)}\\ A=\sqrt{15(15-9)(15-10)(15-11)}\\ A=\sqrt{15(6)(5)(4)}\\ A=\sqrt{1800}\\ A=30\sqrt{2}\\ A=42.426\\ i.e.\\ h_1=\frac{2A}{b_1}=\frac{2A}{a}=\frac{2(42.426)}{9}=9.428\\ h_2=\frac{2A}{b_2}=\frac{2A}{b}=\frac{2(42.426)}{10}=8.485\\ h_3=\frac{2A}{b_3}=\frac{2A}{c}=\frac{2(42.426)}{11}=7.713\end{array}$

Hence,

  $\begin{array}{l}A=42.426sq.unit\\ Heights:9.428,8.485,7.713\end{array}$