In a circle of radius 10cm , A chord is drawn so that it is 4cm distant from the centre. Calculate the length of the minor arc cut off by the chord.
The chord will create a central angle in the circle. To calculate its measure, you can use the cos formula in the right triangle that is formed by the radius, the half-chord, and the segment connecting the center of the circle with the instrument point.
In this case, the radius (r) is 10 cm, and the distance from the center to the chord (d) is 4 cm.
Use the cos formula to find the measure of the angle.
cos(A) = adj/hyp = d/r = 4/10 = 0.4
A = cos^-1(0.4) ≈ 66.42°
Since the chord cuts off two of these angles in the circle, the central angle for the arc is 2 * 66.42° = 132.84°.
The length of an arc (s) on a circle is s = rθ, where θ is the central angle in radians.
To convert the angle to radians, multiply the degree measure by π/180, which yields 132.84° * π/180 = 2.318 radians.
So, the length of the minor arc is s = 10 cm * 2.318 rad ≈ 23.18 cm.
Remember this process gives you the length of the minor arc, which is the shorter arc between the two points where the chord intersects the circle.
To find the length of the minor arc cut off by the chord, we can use the following formula:
Arc Length = θ × r
where θ is the central angle in radians, and r is the radius of the circle.
In this case, we know that the chord is 4cm distant from the center, which means it is 4cm away from the radius of the circle.
Let's draw a line connecting the center of the circle to the chord, creating a right triangle. The radius is one of the legs, and the distance from the center to the chord is the other leg. Since the chord is 4cm distant from the center, the length of the other leg is 4cm.
Now, let's find the length of the hypotenuse using the Pythagorean theorem.
h^2 = r^2 + d^2
where h is the hypotenuse, r is the radius, and d is the distance from the center to the chord.
Substituting the given values, we get:
h^2 = 10^2 + 4^2
h^2 = 100 + 16
h^2 = 116
h = √116
h ≈ 10.77 cm
Now, let's find the central angle θ using the inverse sine function:
sin(θ) = d / h
θ = arcsin(d / h)
Substituting the given values, we get:
θ = arcsin(4 / 10.77)
θ ≈ 0.372 radians
Finally, let's calculate the length of the minor arc:
Arc Length = θ × r
Arc Length ≈ 0.372 × 10
Arc Length ≈ 3.72 cm
Therefore, the length of the minor arc cut off by the chord is approximately 3.72 cm.
To calculate the length of the minor arc cut off by the chord in a circle, we can use the following formula:
Arc Length = Angle / 360° × Circumference
First, let's calculate the angle subtended by the chord at the center of the circle.
We know that the chord is 4 cm distant from the center. So, we can form a right-angled triangle with the radius of the circle, the distance between the chord and the center, and the chord itself.
Using the Pythagorean theorem, we can find the length of the segment connecting the center of the circle to the chord's midpoint:
d^2 = r^2 - h^2
where:
d = distance between the chord and the center
r = radius of the circle
h = distance between the chord and the center
Plugging in the given values:
d^2 = 10^2 - 4^2
d^2 = 100 - 16
d^2 = 84
d ≈ 9.165 cm (rounded to 3 decimal places)
Now, we can find the angle at the center of the circle using trigonometry:
sin(angle/2) = h / r
Plugging in the values:
sin(angle/2) = 4 / 10
angle/2 ≈ sin^(-1)(0.4)
angle/2 ≈ 23.578 degrees (rounded to 3 decimal places)
Therefore, the angle subtended by the chord at the center of the circle is 23.578 degrees.
To calculate the length of the minor arc cut off by the chord, we'll use the formula:
Arc Length = Angle / 360° × Circumference
The circumference of a circle is given by the formula: Circumference = 2πr
Plugging in the values:
Arc Length = (23.578 / 360) × 2π10
Arc Length ≈ (0.065±) × (2 × 3.1416 × 10)
Arc Length ≈ 6.2830 cm (rounded to 4 decimal places)
Therefore, the length of the minor arc cut off by the chord is approximately 6.2830 cm.