there are 4 cirlces in an equilateral triangle with sides equal to 16cm. what is the radius of the big circle and the little circles
To find the radius of the big circle and the small circles in an equilateral triangle, we can use some geometric properties. Here's how:
Let's start with the big circle. Draw three radii of the big circle from its center to the vertices of the equilateral triangle. This forms three smaller congruent equilateral triangles inside the big equilateral triangle.
Now, let's focus on one of the smaller equilateral triangles. The side length of the smaller equilateral triangle can be found by dividing the side length of the big equilateral triangle by 2 (since there are two such triangles on each side). So, the side length of the smaller equilateral triangle is 16 cm / 2 = 8 cm.
Next, we know that in an equilateral triangle, the altitude (or height) divides the triangle into two congruent right triangles. In this case, one side of the right triangle is the radius of the small circle, the other side is half the side length of the small equilateral triangle (4 cm), and the hypotenuse is the radius of the big circle plus the radius of one small circle.
We can use the Pythagorean theorem to find the radius of the small circle. The Pythagorean theorem states that the sum of the squares of the two legs of a right triangle equals the square of the hypotenuse. Hence, we have:
(radius of small circle)^2 + 4^2 = (radius of big circle + radius of small circle)^2
Substituting x for the radius of the small circle, we can solve this equation to find the value of x.
x^2 + 16 = (x + x)^2
x^2 + 16 = 4x^2
3x^2 = 16
x^2 = 16/3
x ≈ √(16/3)
x ≈ 2√(4/3)
x ≈ 2√(1.33)
Therefore, the radius of the small circle is approximately 2√(1.33) cm.
To find the radius of the big circle, we subtract the radius of the small circle from the hypotenuse in the right triangle for the small circle. Using the equation:
(radius of big circle) = (radius of small circle) + (radius of small circle)
(radius of big circle) = 2(2√(1.33))
(radius of big circle) = 4√(1.33)
Thus, the radius of the big circle is approximately 4√(1.33) cm.