an equilateral triangle of side 20cm is inscribed in a circle.calculate the distance of a side of the triangle from th centre of the circle.
5.77cm
The circle center lies at the intersection of the angle bisectors.
Draw a line from the center perpendicular to a side.
The distance from the center to the side is d = 10(sin30º) = 5.
Solutions
Well, isn't this a triangle-tastic question! So, we have an equilateral triangle with each side measuring 20 cm. Let's call the center of the circle "C". Now, to find the distance from a side of the triangle to the center of the circle, we just need to draw a line from the center of the circle to the midpoint of one of the sides of the triangle.
Since the triangle is equilateral, we know that the line from the center to the midpoint is also the altitude of the triangle! Isn't that fabulous? Now, we just need to find the length of this altitude.
To calculate the altitude of an equilateral triangle, we can use a handy-dandy formula:
altitude = (s√3) / 2
Where "s" is the length of the side of the triangle. Plugging in our values, we get:
altitude = (20√3) / 2
Calculating this, we find that the altitude of the triangle is approximately 17.32 cm. So, the distance from a side of the triangle to the center of the circle is 17.32 cm.
Voila! Now you know how to keep your triangles and circles in perfect balance!
To calculate the distance from the center of the circle to a side of the equilateral triangle, we can make use of some geometric properties.
In an equilateral triangle, all three sides are equal, and each angle is 60 degrees. When the triangle is inscribed in a circle, each side of the triangle touches the circle at a point called the tangent point.
By drawing a line segment from the center of the circle to one of the tangent points, we create a right triangle. The distance from the center of the circle to the side of the triangle is equal to the length of the altitude of this right triangle.
To calculate the altitude, we can use the Pythagorean Theorem, which states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
Let's denote the distance from the center of the circle to the side of the triangle as 'd'. The hypotenuse of the right triangle is the radius of the circle, which is equal to half the length of a side of the triangle (since the triangle is inscribed in the circle).
Given that the side of the equilateral triangle is 20 cm, the radius of the circle is 10 cm.
Using the Pythagorean Theorem:
d^2 + 10^2 = 20^2
d^2 + 100 = 400
d^2 = 300
Taking the square root of both sides, we find:
d ≈ 17.32 cm
Therefore, the distance from the center of the circle to a side of the equilateral triangle is approximately 17.32 cm.