A chord is 4cm from the centre of a circle of radius 7cm. Calculate (a)the angle subtended by the chord at the centre (b) the length of the minnor arc cut off by the chord
Did you make your sketch?
let Ø be half the central angle, or the angle at the centre for the triangle
cosØ = 4/7
Ø = appr 55.15°
so the central angle is 110.30°
circumference = 2πr = 14π
minor arc/14π = 110.3/360
minor arc = ....
Please solution is not complete
To find the angle subtended by a chord at the center of a circle, you can use the relationship between the chord length and the radius of the circle.
(a) To calculate the angle subtended by the chord at the center of the circle:
1. First, find the length of the chord. In this case, the chord is 4 cm from the center, so it cuts the circle into two segments.
2. To find the length of the chord, we can derive a right-angled triangle using the radius and the distance from the chord to the center. The radius is 7 cm, and the distance from the chord to the center is half of the chord length.
Using the Pythagorean theorem, we can solve for the chord length:
chord length = 2 * √(radius^2 - distance^2)
= 2 * √(7^2 - 2^2)
= 2 * √(49 - 4)
= 2 * √(45)
= 2 * 3√5
= 6√5 cm
3. The angle subtended by the chord at the center can be calculated using the formula:
angle = 2 * arcsin(chord length / (2 * radius))
= 2 * arcsin((6√5) / (2 * 7))
= 2 * arcsin(3√5 / 7)
(b) To calculate the length of the minor arc cut off by the chord:
1. The length of the minor arc cut off by the chord is simply the circumference of the sector formed by the chord.
2. The circumference of the sector can be calculated using the formula:
arc length = (angle / 360 degrees) * 2 * π * radius
= (angle / 360) * 2 * π * 7 cm
So, the length of the minor arc cut off by the chord is given by (angle / 360) * 2 * π * 7 cm.