A traffic signal board, indicating 'SCHOOL AHEAD', is an equilateral triangle with side 'a'. Find the area of the signal board, using Heron's formula. If its perimeter is 180 cm, w... A traffic signal board, indicating 'SCHOOL AHEAD', is an equilateral triangle with side 'a'. Find the area of the signal board, using Heron's formula. If its perimeter is 180 cm, what will be the area of the signal board?
Understand the Problem
The image contains multiple geometry problems related to finding the area of triangles. The questions involve using Heron's formula, understanding ratios, and applying the properties of equilateral and isosceles triangles. We need to extract and classify each question separately.
Answer
$900\sqrt{3} \text{ cm}^2$
Answer for screen readers
$900\sqrt{3} \text{ cm}^2$
Steps to Solve
- Find the semi-perimeter s The semi-perimeter is half of the perimeter of the triangle. Given the perimeter is 180 cm, we calculate $s$.
$ s = \frac{180}{2} = 90 $
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Apply Heron's Formula for an equilateral triangle Heron's formula is given by $\sqrt{s(s-a)(s-b)(s-c)}$. Since it is an equilateral triangle, $a = b = c$. Also, since the perimeter is 180 cm, each side $a = \frac{180}{3} = 60$ cm.
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Substitute values into Heron's Formula We substitute $s = 90$ and $a = b = c = 60$ into Heron's formula:
Area $ = \sqrt{90(90-60)(90-60)(90-60)} $ $ = \sqrt{90(30)(30)(30)} $ $ = \sqrt{90 \times 27000} $ $ = \sqrt{2430000} $ $ = \sqrt{810000 \times 3} $ $ = 900\sqrt{3} $
- State the final area The area of the signal board is $900\sqrt{3}$ cm$^2$.
$900\sqrt{3} \text{ cm}^2$
More Information
Heron's formula is a useful tool to find the area of a triangle when you know the length of all three sides. In the special case of an equilateral triangle, all sides are equal ($a=b=c$), which simplifies the calculations.
Tips
A common mistake is not simplifying the square root correctly. Make sure to factor out perfect squares before taking the square root. Also, one might miscalculate the semi-perimeter or the length of the side of the triangle.
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