In a circle , the radius is 68 units and a chord of the circle is 120 units.
Part A)How far is the chord from the center?
Part B)Find the measure of the arc created by the chord.
A) Let x be the distance of the chord from the center. Draw a line from the center of the chord to the center of the circle. This forms a right triangle with a radius and half the chord.
x^2 + 60^2 = r^2 = 4624
x^2 = 1024
x = 32
B) The angle subtended by the chord is 2 arcsin 60/68 = 123.85 degrees = 2.162 radians
Multiply that by the radius to get the arc length, 147.0 units.. a bit longer than the chord length, as you would expect.
Part A) To find the distance of the chord from the center of the circle, we need to draw perpendiculars from the center of the circle to the chord. This will create right triangles.
Let's call the center of the circle point O, the midpoint of the chord point M, and the distance from O to M point D. We can use the Pythagorean theorem to find the distance OD.
In a right triangle, the hypotenuse is the radius of the circle (68 units), one leg is half the length of the chord (120/2 = 60 units), and the other leg is the distance from the center to the chord (OD).
Using the Pythagorean theorem, we can write:
OD^2 + 60^2 = 68^2
OD^2 + 3600 = 4624
OD^2 = 4624 - 3600
OD^2 = 1024
OD = sqrt(1024)
OD = 32 units
So the chord is 32 units away from the center of the circle.
Part B) To find the measure of the arc created by the chord, we need to find the angle formed by the endpoints of the chord and the center of the circle.
To find this angle, we can use the properties of a central angle. The measure of an arc is equal to the measure of its central angle. So we need to find the central angle that intercepts the arc created by the chord.
In a right triangle, the length of the chord is the base, and the distance from the center to the chord is the height. The central angle is formed at the center of the circle, so it intercepts the arc created by the chord.
Using trigonometry, we can write:
sin(theta) = opposite/hypotenuse
sin(theta) = 60/68
theta = arcsin(60/68)
Using a calculator, we can find:
theta ≈ 57.78 degrees
So the measure of the arc created by the chord is approximately 57.78 degrees.
To answer both Part A and Part B, we can use the properties of a circle to find the solutions.
Part A: How far is the chord from the center?
To determine the distance between the chord and the center of the circle, we can use the following steps:
1. Draw a line connecting the center of the circle to the midpoint of the chord. This line is called the perpendicular bisector of the chord.
2. Since the line from the center to the midpoint of the chord is perpendicular to the chord, it divides the chord into two equal parts.
3. Therefore, we have two right triangles formed, each with the radius of the circle as the hypotenuse (68 units) and half the length of the chord (120/2 = 60 units) as the base.
4. Using the Pythagorean theorem, we can find the height (distance from the center to the chord).
a^2 + b^2 = c^2
Where a and b are the legs of the right triangle, and c is the hypotenuse.
In this case, we have:
a^2 + 60^2 = 68^2
a^2 + 3600 = 4624
a^2 = 4624 - 3600
a^2 = 1024
Taking the square root of both sides gives us:
a = √1024
a = 32
Therefore, the distance between the chord and the center of the circle is 32 units.
Part B: Find the measure of the arc created by the chord.
The measure of the arc created by the chord can be calculated using the formula:
θ = 2 * arcsin(b / 2r)
Where θ is the measure of the arc, b is the length of the chord, and r is the radius of the circle.
In this case, we have:
θ = 2 * arcsin(120 / (2 * 68))
θ = 2 * arcsin(0.882)
Using a scientific calculator or trigonometric tables, we can determine the arcsin value:
θ = 2 * 61.07°
θ = 122.14°
Therefore, the measure of the arc created by the chord is approximately 122.14 degrees.