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.
To find the length of the minor arc cut off by the chord, we need to find the angle subtended by that chord at the center of the circle.
Let's denote the center of the circle as O and the end points of the chord as P and Q. The chord PQ cuts off two arcs on the circle, a major arc and a minor arc. We are asked to find the length of the minor arc.
Since O is the center of the circle and OP and OQ are radii, triangle OPQ is an isosceles triangle with OP = OQ = 10 cm.
Also, the perpendicular distance from the center O to chord PQ is given as 4 cm. Let's denote it as OM.
In the right triangle OMP, using the Pythagorean Theorem, we can find the length of the half chord MP:
MP = √(OP² - OM²) = √(10² - 4²) = √(100 - 16) = √84 cm.
So the whole length of the chord PQ = 2 * MP = 2 * √84 cm.
In triangle OMP, sin(∠OMP) = OM / OP = 4 / 10 = 0.4
So, ∠OMP = arcsin(0.4) = 23.58 degrees.
Because ∠OMP is half the angle subtended by the chord at the center, the whole angle is 2 * ∠OMP = 2 * 23.58 = 47.16 degrees.
Because the ratio of the length of an arc to the circumference of the whole circle is equal to the ratio of the angle it subtends at the center to 360 degrees:
Length of the minor arc = (47.16 / 360) * Circumference
= (47.16 / 360) * 2πr
= (47.16 / 360) * 2π * 10
= 26.13 cm.
This is the length of the minor arc cut off by the chord.
To find the length of the minor arc cut off by the chord, we need to calculate the angle at the center of the circle that the chord subtends. We can use the properties of a right triangle to find this angle.
Let's label the following points:
- The center of the circle as O
- The midpoint of the chord as M
- The point where the chord intersects the circle as A
According to the information given, OA is the radius of the circle, which is 10 cm. AM is 4 cm since the chord is 4 cm distant from the center.
Since triangle OMA is a right triangle, OM is the hypotenuse. We can use the Pythagorean theorem to find OM.
OM^2 = OA^2 - AM^2
OM^2 = 10^2 - 4^2
OM^2 = 100 - 16
OM^2 = 84
Taking the square root of both sides, we get:
OM = √84
OM ≈ 9.165 cm (rounded to 3 decimal places)
Now, we know that OA is the radius and OM is the hypotenuse of right triangle OMA. We can use trigonometric ratios to find the angle θ at the center of the circle that the chord subtends.
sin(θ) = AM / OM
sin(θ) = 4 / 9.165
θ = arcsin(4 / 9.165)
θ ≈ 25.62 degrees (rounded to 2 decimal places)
The length of the minor arc cut off by the chord can be calculated using the formula:
Arc length = (θ / 360) * 2πr
Plugging in the values, we have:
Arc length = (25.62 / 360) * 2π * 10
Arc length ≈ (0.0712) * (2 * 3.1416 * 10)
Arc length ≈ 0.1424 * 62.832
Arc length ≈ 8.93 cm (rounded to 2 decimal places)
Therefore, the length of the minor arc cut off by the chord is approximately 8.93 cm.
To calculate the length of the minor arc cut off by the chord in a circle, you can use the following formula:
\(\text{Arc length} = \text{Radius} \times \theta\)
Where:
- \(\text{Arc length}\) is the length of the minor arc
- \(\text{Radius}\) is the radius of the circle
- \(\theta\) is the central angle (in radians) subtended by the arc at the center of the circle
To find the central angle, we can use trigonometry. Since the chord is 4cm distant from the center, it forms a right triangle with the radius and the distance between the chord and the center. This distance is also known as the perpendicular distance from the center to the chord.
Using the Pythagorean theorem, we can find the length of the perpendicular distance:
\(r^2 = 4^2 + 10^2\) (where \(r\) is the perpendicular distance)
\(r^2 = 116\)
\(r = \sqrt{116}\)
\(r \approx 10.77 cm\)
Now, we can use trigonometry to find the central angle \(\theta\). Since we have a right triangle with the adjacent side (radius) of 10cm and the hypotenuse (perpendicular distance) of 10.77 cm, we can use the cosine function:
\(\cos(\theta) = \frac{{\text{Adjacent}}}{{\text{Hypotenuse}}} = \frac{{10}}{{10.77}}\)
\(\theta = \cos^{-1}(\frac{{10}}{{10.77}})\)
Using a calculator, you can calculate \(\theta\) to be approximately 0.869 radians.
Finally, we can calculate the length of the minor arc:
\(\text{Arc length} = \text{Radius} \times \theta = 10 \times 0.869\)
\(\text{Arc length} \approx 8.69 cm\)
Therefore, the length of the minor arc cut off by the chord is approximately 8.69 cm.