Answer:
One side of the polygon = 51.96 m
Perimeter = 155.88 m
Area of the polygon = 1169.13 m²
Step-by-step explanation:
The given diagram shows a 3-sided regular polygon (equilateral triangle) with an apothem that measures 15 m.
[tex]\dotfill[/tex]
Side Length
The apothem of an equilateral triangle represents the shortest leg of a 30-60-90 right triangle, which means that the longest leg of the 30-60-90 triangle is equal to half the length of the polygon's side.
The sides of a 30-60-90 triangle are in the ratio 1 : √3 : 2, so the longest leg is √3 times the length of the shortest leg. Given that the shortest leg of the 30-60-90 triangle is 15 m (apothem), then the longest leg is 15√3 m. As the side length the polygon is twice this, then:
[tex]\textsf{One side of the polygon}=\sf Apothem \times \sqrt{3} \times 2\\\\\textsf{One side of the polygon}=15\sqrt{3} \times 2\\\\\textsf{One side of the polygon}=30\sqrt{3}\\\\\textsf{One side of the polygon}=51.96\; \sf m\;(nearest\;hundredth)[/tex]
Therefore, one side of the polygon measures 51.96 m.
[tex]\dotfill[/tex]
Perimeter
The perimeter of a regular polygon can be found by multiplying the length of one side by the number of sides. Therefore:
[tex]\textsf{Perimeter}=\sf Side \;length \times Number\;of\;sides\\\\\textsf{Perimeter}=30\sqrt{3} \times 3\\\\\textsf{Perimeter}=90\sqrt{3}\\\\\textsf{Perimeter}=155.88\; \sf m\;(nearest\;hundredth)[/tex]
Therefore, the perimeter of the polygon is 155.88 m.
[tex]\dotfill[/tex]
Area
The area of a regular polygon is half the product of its perimeter and apothem. So:
[tex]\sf \textsf{Area}=\dfrac{\sf Perimeter \times Apothem}{2}\\\\\\\textsf{Area}=\dfrac{90\sqrt{3} \times 15}{2}\\\\\\\textsf{Area}=675\sqrt{3}\\\\\\\textsf{Area}=1169.13\; \sf m^2\;(nearest\;hundredth)[/tex]
Therefore, the area of the polygon is 1169.13 m².
