a belt is wrapped tightly around circle O and forms a right angle at P, outside the circle. find the length of the belt if circle O has a radius of 8.
To find the length of the belt, we need to determine the distance covered along the circumference of the circle.
Let's start by determining the distance covered by the belt along the two sides of the right triangle formed. We know that one side of the triangle is the radius of the circle, which is 8 units.
The length of the hypotenuse of the triangle is equal to the length of the belt. To find this length, we need to determine the length of the other side of the triangle.
Since the triangle is a right triangle and one of the angles is 90 degrees, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b): a^2 + b^2 = c^2.
In this case, one of the sides (a) is the radius of the circle (8 units), and the other side (b) is the distance covered by the belt along the circle's circumference. Let's call this distance x.
Applying the Pythagorean theorem, we get:
8^2 + x^2 = c^2
64 + x^2 = c^2
Now, we need to find the value of x.
Since the belt wraps tightly around the circle, the distance covered along the circle's circumference is equal to the arc length between points P and the point where the belt comes in contact with the circle, which we named Q.
The angle formed at the center of the circle between points P and Q is 90 degrees. An angle at the center of a circle is twice the angle formed at any point on the circumference of the circle by the same arc.
Therefore, the angle formed at P by the arc PQ is 90/2 = 45 degrees.
The length of the arc PQ can be found using the formula for the length of an arc: Length = (angle/360) * 2 * π * r, where r is the radius of the circle.
Substituting the values, we get:
Length of arc PQ = (45/360) * 2 * π * 8 = (1/8) * 16π = 2π units
This length represents the distance covered by the belt along the circle's circumference (b in the Pythagorean theorem).
Now we can substitute this value into the previously derived equation:
64 + x^2 = c^2
64 + (2π)^2 = c^2
64 + 4π^2 = c^2
To find the value of c (the length of the belt), we need to take the square root of both sides:
√(64 + 4π^2) = c
Calculating the square root, we find:
c ≈ 19.5 units
Therefore, the length of the belt is approximately 19.5 units.