an equilateral triangle of side 20cm is inscribed in a circle calculate the distance of a side of the triangle from the centre of the circle
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As you know, the intersections of the medians is 2/3 of the way from each vertex to the opposite side.
So, the radius of the circle is 2/3 the length of the altitudes (which are also medians for equilateral triangles.
The altitude of an equilateral triangle of side s is s/2 √3
So, the distance from the center to the side is 1/3 * 20/2 √3 = 10/√3
To find the distance of a side of the triangle from the center of the circle, we can use some properties of an equilateral triangle inscribed in a circle.
In an equilateral triangle inscribed in a circle, the center of the circle coincides with the centroid of the triangle, which is also the point of concurrency of the triangle's medians.
The centroid of an equilateral triangle divides each median into segments in a 2:1 ratio, with the longer segment being closer to the centroid.
Since the centroid is equidistant from the three vertices of the triangle, the distance from the center of the circle to any side of the triangle is equal to 2/3 times the radius of the circle.
In this case, since the side of the triangle is 20 cm, the radius of the circle can be calculated as follows:
The median divides the equilateral triangle into two congruent right-angled triangles, each with the hypotenuse being the side of the triangle (20 cm) and one of the legs being half the side of the triangle (10 cm).
Using the Pythagorean theorem, we can find the length of the other leg, which is the height of the triangle:
height^2 = hypotenuse^2 - leg^2
height^2 = 20^2 - 10^2
height^2 = 400 - 100
height^2 = 300
height = √300 cm
height ≈ 17.32 cm
Since the height is also the distance from the center of the circle to any side of the triangle, the distance from the center of the circle to a side of the triangle is 2/3 times the height:
Distance = (2/3) * height
Distance = (2/3) * 17.32 cm
Distance ≈ 11.55 cm
Therefore, the distance of a side of the equilateral triangle from the center of the circle is approximately 11.55 cm.
To find the distance of a side of the equilateral triangle from the center of the circle, we can use a property of equilateral triangles. In an equilateral triangle, the distance from any side to the center is equal to the radius of the circumscribing circle.
The radius of the circumscribing circle can be found using the formula:
radius = (side length / √3)
In this case, the side length of the equilateral triangle is given as 20 cm. So, we can substitute this value into the formula to find the radius:
radius = (20 cm / √3)
To get an exact value, we can simplify the expression.
Simplifying the expression involves rationalizing the denominator:
radius = (20 cm / √3) * (√3 / √3)
= (20√3 cm) / 3
Therefore, the distance of a side of the equilateral triangle from the center of the circle is (20√3) / 3 cm.