Answer:
area ≈ 24π
Step-by-step explanation:
We have solved this problem two ways:
- Using a drawing program that writes the formula of the circumscribing circle, so gives the value of r^2.
- Using rarely-seen formulas for the area of a triangle and for the area of its circumscribing circle.
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Drawing
A drawing of the figure (below) can help you find the radius of circle O. It is about 4.89 inches, so the area of circle O is about ...
area = πr^2 = π(4.89 in)^2 ≈ 23.9π ≈ 24π . . . .square inches
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Formulas
There is an interesting relationship between the area of the triangle and the radius of the circumscribing circle:
r = (abc)/(4A) . . . . . where a, b, c are the triangle side lengths, and A is the triangle area
Heron's formula can tell us the area of the triangle from the side lengths:
A = √(s(s-a)(s-b)(s-c)) . . . . where s = (a+b+c)/2
For the given triangle with side lengths 4, 5, and 8 (inches), the area can be found as ...
s = (4+5+8)/2 = 8.5
A = √(8.5·4.5·3.5·0.5) = √66.9375 ≈ 8.1815 . . . square inches
Then the radius of the circle is ...
r = (4·5·8)/(4·8.1815) = 4.889 . . . inches
The area of the circle is then ...
Circle O area = πr^2 = π(4.889 in)^2 = 23.9π in^2
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The closest answer choice is 3.14×22.
