Answer:
- radius: 32
- side length: 32√3
- triangle area: 768√3
- inscribed circle area: 256π
- circumscribed circle area: 1024π
Explanation:
For geometry problems, it is usually helpful to start by making a diagram. In the attached diagram, you are given equilateral triangle ABC and the apothem OX = 16.
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1. Triangle Radius
The length OA is the radius of the triangle (the radius of the circumcircle). Its relation to the apothem is ...
OX/OA = sin(30°) = 1/2
OA = 2·OX = 2·16 = 32
The radius of the regular triangle is 32.
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2. Triangle side length
The length AX is half the side length. Its relation to AX is ...
OX/AX = tan(30°) = (√3)/3
AX = OX·√3 = 16√3
AC = 2·AX = 32√3
The side length of the regular triangle is 32√3.
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3. Triangle area
The area formula can be used. We know the side length from part 2. and we know the height of the triangle, BX, is the sum of the radius and the apothem, so is 32+16 = 48.
Triangle Area = (1/2)bh = (1/2)(32√3)(48) = 768√3
The area of the triangle is 768√3.
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4. Area of Inscribed circle
The radius of the inscribed circle is the apothem, 16. So the circle area is ...
A = πr² = π·16² = 256π
The area of the inscribed circle is 256π.
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5. Area of Circumscribed circle
The radius of the circumscribed circle is found in part 1 as 32. So the circle area is ...
A = πr² = π·32² = 1024π
The area of the circumscribed circle is 1024π.