An equilateral triangle of side 20cm is inscribed in a circle. Calculate the distance of a side of the triangle from the center of the circle.
the center of the triangle is 2/3 of the distance from a vertex to the opposite side (2/3 of the height of the triangle)
so the distance from the center of the circle to a side of the triangle is 1/3 the height of the triangle
To calculate the distance of a side of the equilateral triangle from the center of the circle, we need to find the length of the perpendicular from the center of the circle to any side of the triangle.
In an equilateral triangle, the center of the circle is also the circumcenter, which means it is equidistant from all three vertices of the triangle. Hence, the line segment connecting the center of the circle to any side of the triangle is a perpendicular bisector.
Let's label the triangle ABC, with the center of the circle O, and D as the point where the perpendicular from O meets side AB.
Now, we have an isosceles right triangle AOD, with AO as the hypotenuse and AD as each leg. We know that the length of one side of the equilateral triangle is 20 cm, so the length of AD is half of that, which is 10 cm.
To find the length of AO, we can use the Pythagorean theorem:
AO^2 = AD^2 + OD^2
Since AD is 10 cm, we need to find OD. Since D is the midpoint of side AB (due to the perpendicular bisector), we can find OD using the Pythagorean theorem again:
OD^2 = AB^2 - AD^2
OD^2 = 20^2 - 10^2
OD^2 = 400 - 100
OD^2 = 300
Now, substitute the value of OD^2 into the first equation:
AO^2 = AD^2 + OD^2
AO^2 = 10^2 + 300
AO^2 = 100 + 300
AO^2 = 400
Taking the square root of both sides, we get:
AO = √400
AO = 20 cm
Therefore, the distance of a side of the equilateral triangle from the center of the circle is 20 cm.