Chapter 1.2, Problem 2E

(a)

To determine

To find:The formula for height $h$ of an equilateral triangle as a function of its side length $s$ .

Answer to Problem 2E

The formula of theheight $h$ of an equilateral triangle as a function of its side length $s$ is $h=\frac{\sqrt{3}}{2}s$ .

Explanation of Solution

Given information: Theheight of an equilateral triangle is $h$ and side length is $s$ .

Formulaused:Pythagoras theorem - In a right angle triangle, the sum of its square of its base and height is equal to the square of the hypotenuse.

  ${\left(\text{hypotenuse}\right)}^2={\left(\text{base}\right)}^2+{\left(\text{height}\right)}^2$

Calculation:

The height of an equilateral triangle is the perpendicular line drawn from a point to its base. So, the height is the median of the triangle.

The length of the base is $\frac{s}{2}$ and the length of hypotenuse is $s$ .

Substitute $\frac{s}{2}$ for base and $s$ for hypotenuse in the Pythagoras formula as:

  $\begin{array}{c}{h}^2+{\left(\frac{s}{2}\right)}^2=s^2\\ h^2=s^2-\frac{s^2}{4}\\ h^2=\frac{3s^2}{4}\\ h=\frac{\sqrt{3}}{2}s\end{array}$

Therefore, the formula of the height $h$ of an equilateral triangle as a function of its side length $s$ is $h=\frac{\sqrt{3}}{2}s$ .

(b)

To determine

To find:The height of an equilateral triangle with side length $3\text{m}$ .

Answer to Problem 2E

The height of an equilateral triangle with side length $3\text{m}$ is $1.5\sqrt{3}\text{m}$ .

Explanation of Solution

Given information:The length of the side of an equilateral triangle is $3\text{m}$ .

Calculation:

As calculated in part(a), the formula for height of an equilateral triangle as a function of its side length $s$ is:

  $h=\frac{\sqrt{3}}{2}s$

Substitute $3$ for $s$ in the above formula.

  $\begin{array}{l}h=\frac{\sqrt{3}}{2}s\\ h=\frac{\sqrt{3}}{2}\times \left(3\text{m}\right)\\ h=1.5\sqrt{3}\text{m}\end{array}$

Therefore, the height of an equilateral triangle with side length $3\text{m}$ is $1.5\sqrt{3}\text{m}$ .