Question

bianca calculated the height of the equilateral triangle with side lengths of 10
\\( \tan ( 30 ) = \frac { 5 } { h } \\)
\\( h \approx 8.7 \\)
then, she used the formula for area of a triangle to approximate its area, as shown below.
\\( \

$$\begin{array} { r l } { a } & { = \\frac { 1 } { 2 } b h } \\\\ { } & { = \\frac { 1 } { 2 } ( 10 ) ( 8.7 ) } \\\\ { } & { = 43.5 \\text { units } ^ { 2 } } \\end{array}$$

\\)
calculate the area of the equilateral triangle using the formula for area of a regular polygon, and compare it to biancas answer
the apothem, rounded to the nearest tenth, is units.
the perimeter of the equilateral triangle is units
therefore, the area of the equilateral triangle is or approximately 43.5 units ^ { 2 }
the calculated areas are

Explanation:

Step1: Calculate the apothem

For an equilateral triangle, the formula for the apothem \(a\) (using the relationship in a \(30 - 60-90\) triangle related to the center of the triangle) can also be derived from the formula \(A=\frac{1}{2}aP\) (area of a regular polygon). Another way: In an equilateral triangle with side length \(s = 10\), the formula for the apothem \(a\) is \(a=\frac{\sqrt{3}}{6}s\). But if we use the formula \(A=\frac{1}{2}aP\) and we know from Bianca's calculation \(A = 43.5\) and \(P=3s=3\times10 = 30\). Rearranging \(A=\frac{1}{2}aP\) for \(a\), we get \(a=\frac{2A}{P}\). Substituting \(A = 43.5\) and \(P = 30\), we have \(a=\frac{2\times43.5}{30}=2.9\)

Step2: Calculate the perimeter

The perimeter \(P\) of an equilateral triangle with side length \(s\) is \(P = 3s\). Given \(s=10\), then \(P=3\times10=30\)

Step3: Calculate the area using the regular - polygon formula

Using the formula for the area of a regular polygon \(A=\frac{1}{2}aP\), with \(a = 2.9\) and \(P=30\), we get \(A=\frac{1}{2}\times2.9\times30=43.5\)

Answer:

The apothem, rounded to the nearest tenth, is \(2.9\) units.
The perimeter of the equilateral triangle is \(30\) units.
Therefore, the area of the equilateral triangle is \(\frac{1}{2}\times2.9\times30\) or approximately \(43.5\) units².
The calculated areas are the same.