6. The circle has a circumference of 345cm. Line AC is the diameter of the circle. Square ABCD is made up of 4 equilateral triangles. What is the side length of side AB. Show your work for full marks. Remember. The image is not to scale.
To find the side length of side AB, we first need to find the radius of the circle.
The formula for the circumference of a circle is C = 2πr, where C is the circumference and r is the radius.
In this case, we have a circumference of 345 cm, so we can set up the equation as follows:
345 = 2πr
To solve for r, we divide both sides of the equation by 2π:
r = 345 / (2π)
Next, we know that line AC is the diameter of the circle, which means it passes through the center of the circle. Since the square ABCD is made up of 4 equilateral triangles, and an equilateral triangle has all sides equal, we can conclude that line AC is the side length of each equilateral triangle.
Therefore, the side length of each equilateral triangle (and also side AB of the square) is equal to twice the radius. We can calculate this by multiplying the radius by 2:
2 * r = 2 * (345 / (2π))
Simplifying further:
2 * r = 345 / π
Thus, the side length of side AB is equal to 345 divided by π, which is approximately 109.901 cm.
To find the side length of side AB in square ABCD, we can use the information given about the circle's circumference.
First, let's find the diameter of the circle, which is line AC. The circumference of a circle can be calculated using the formula: C = π * d, where C is the circumference and d is the diameter.
Given that the circumference of the circle is 345 cm, we can substitute this value into the formula to find the diameter:
345 cm = π * d
To solve for d, divide both sides of the equation by π:
d = 345 cm / π
Next, let's find the side length of the equilateral triangles in the square. In an equilateral triangle, all sides are equal.
In square ABCD, each side of the square is equal to the side length of the equilateral triangle. To find the side length of the square, we can divide the diameter of the circle by √3, because the side length of an equilateral triangle is equal to the diameter divided by √3.
The side length of side AB = d / √3, where d is the diameter of the circle.
Substituting the value of the diameter we found earlier:
Side length of side AB = (345 cm / π) / √3
So, the side length of side AB is (345 cm / π) / √3 cm.
To find the side length of side AB in the square ABCD, we first need to find the diameter of the circle, which is line AC.
Given that the circumference of the circle is 345 cm, we can use the formula for circumference: circumference = π × diameter.
So, we have the equation 345 cm = π × AC.
To find AC, divide both sides of the equation by π: AC = (345 cm) / π.
Now, let's find the side length of the equilateral triangles in square ABCD.
Since the square is made up of 4 equilateral triangles, each triangle will have 1/4th of the total area of the square.
Since the triangles are equilateral, all sides have the same length.
Let's denote the side length of AB (which is equal to the side length of each triangle) as x.
The area of the square ABCD is equal to the sum of the areas of the 4 equilateral triangles, which is given by the formula: area of square = 4 × area of triangle.
The formula for the area of an equilateral triangle is: area of triangle = (sqrt(3) / 4) × side length^2.
Substituting the side length x into the area formula, we have: area of triangle = (sqrt(3) / 4) × x^2.
Since the area of the square is also equal to the sum of the areas of the 4 triangles, we have: area of square = 4 × [(sqrt(3) / 4) × x^2].
Given that the square ABCD has 4 equal sides, the area of the square is equal to side length^2, so we can write: x^2 = 4 × [(sqrt(3) / 4) × x^2].
Dividing both sides of the equation by 4, we get: x^2 = (sqrt(3) / 4) × x^2.
Cancelling out the common factor of x^2, we have: 1 = sqrt(3) / 4.
To isolate x (the side length of the triangle), we divide both sides by sqrt(3) / 4:
x = 4 / sqrt(3).
Thus, the side length of side AB is 4 / sqrt(3) cm.