29. An equilateral triangle of side 10cm is inscribed in a circle. Find the radius of the circle
the altitude of the triangle is 5√3
The altitudes are also medians.
The medians intersect 2/3 of the way from a vertex to the side. That is also the center of the circle.
So, r = 2/3 * 5√3 = 10/√3
To find the radius of the circle inscribed in an equilateral triangle, we can use the following formula:
r = (s*√3)/6
where r is the radius of the circle and s is the length of a side of the equilateral triangle.
Given that the side length of the equilateral triangle is 10 cm, we can substitute this value into the formula to find the radius:
r = (10 * √3) / 6
Simplifying this expression, we get:
r = 10√3 / 6
To simplify further, we can divide both the numerator and the denominator by 2:
r = (5√3) / 3
Therefore, the radius of the circle inscribed in the equilateral triangle with a side length of 10 cm is (5√3)/3.
To find the radius of the circle inscribed in an equilateral triangle, you can use the following steps:
Step 1: Draw the equilateral triangle and the circle inscribed in it.
Step 2: In an equilateral triangle, all three sides are equal. Since the side of the triangle is given as 10cm, the lengths of all three sides are 10cm.
Step 3: Draw a perpendicular line from the center of the circle to one of the sides of the triangle. This line bisects the side of the triangle and touches the side at a right angle.
Step 4: This perpendicular line divides the equilateral triangle into two congruent right triangles. The length of each shorter side of the right triangle is half the length of the side of the equilateral triangle, which is 10cm/2 = 5cm.
Step 5: Now we have a right triangle with one side of length 5cm and the hypotenuse equal to the radius of the circle. We can use the Pythagorean theorem to find the length of the hypotenuse (radius).
Step 6: By applying the Pythagorean theorem, we have:
radius^2 = (5cm)^2 + (10cm/2)^2
= 25cm^2 + 25cm^2
= 50cm^2
Step 7: Taking the square root of both sides, we have:
radius = √50cm^2
≈ 7.07cm
Therefore, the radius of the circle inscribed in the equilateral triangle is approximately 7.07cm.