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 the belt covers along the circumference of the circle.
First, let's analyze the given information. We have a circle, O, with a radius of 8 units. The belt is wrapped tightly around the circle and forms a right angle at point P, outside the circle.
Since the belt is wrapped tightly around the circle, the segment from point P to the circumference of the circle (which we will call point Q) is perpendicular to the radius OP.
We can use the Pythagorean Theorem to find the length of line segment PQ, which represents the distance the belt covers along the circumference of the circle.
According to the Pythagorean Theorem, the square of the length of the hypotenuse (PQ) is equal to the sum of the squares of the other two sides (OP and OQ) of a right triangle.
In this case, OP is the radius of the circle, which is given as 8 units. We need to find OQ.
Since OP and OQ are sides of a right triangle, we can use the Pythagorean Theorem.
OP^2 + OQ^2 = PQ^2
8^2 + OQ^2 = PQ^2
64 + OQ^2 = PQ^2
We know that OQ = radius = 8, so let's substitute this value into the equation.
64 + 8^2 = PQ^2
64 + 64 = PQ^2
128 = PQ^2
Now, take the square root of both sides to find the length of PQ.
√128 = √PQ^2
√(2^7) = PQ
2√(2^6) = PQ
2 * 2^3 = PQ
16 = PQ
Therefore, PQ, which represents the length of belt along the circumference of the circle, is 16 units.
To find the total length of the belt, we need to consider that the belt wraps around the circle once completely, covering the entire circumference. The circumference of a circle is given by the formula:
Circumference = 2πr
Substituting the given radius value of 8, we can calculate the circumference.
Circumference = 2π(8)
Circumference = 16π
So, the total length of the belt is 16π units.