In a circle of radius 6cm. A chord is drawn 3cm from the centre
a. Calculate the angle subtended by the chord at the centre of the circle
b. Hence, find the length of the minor arc cut off the chord.
a. The chord is 3cm from the center, so it divides the radius into two parts: a segment of length 3cm and a segment of length 6cm - 3cm = 3cm. These segments, the chord, and the center of the circle form an isosceles triangle.
Using the Pythagorean theorem, we can find the length of the altitude of this triangle (which is also the perpendicular bisector of the chord):
$$(6\text{ cm})^2 - (3\text{ cm})^2 = 36 - 9 = 27$$
So the length of the altitude is $\sqrt{27} = 3\sqrt{3}$ cm.
Now we can find the angle $\theta$ subtended by the chord at the center of the circle, using the formula for the area of a triangle:
$$A = \frac{1}{2}bh$$
where $b$ is the length of the chord and $h$ is the length of the altitude. In this case, $b = 6$ cm and $h = 3\sqrt{3}$ cm, so:
$$A = \frac{1}{2}(6\text{ cm})(3\sqrt{3}\text{ cm}) = 9\sqrt{3}\text{ cm}^2$$
This is also the area of the sector of the circle subtended by the angle $\theta$. The area of a sector is given by:
$$A = \frac{1}{2}r^2\theta$$
where $r$ is the radius of the circle. In this case, $r = 6$ cm, so we have:
$$9\sqrt{3}\text{ cm}^2 = \frac{1}{2}(6\text{ cm})^2\theta$$
Simplifying:
$$\theta = \frac{9\sqrt{3}}{6^2}\text{ rad} = \frac{\sqrt{3}}{4}\text{ rad}$$
To convert to degrees, we can use the fact that $2\pi$ radians equals 360 degrees, so:
$$\theta = \frac{\sqrt{3}}{4}\text{ rad} \approx 15.52^\circ$$
b. The minor arc cut off by the chord is simply the part of the circle bounded by the chord. This is a segment of the circle, so we can find its length using the formula for the length of a segment:
$$L = r\theta$$
where $r$ is the radius of the circle and $\theta$ is the angle subtended by the segment at the center of the circle. In this case, $r = 6$ cm and $\theta$ is the angle we calculated in part a:
$$L = (6\text{ cm})\left(\frac{\sqrt{3}}{4}\text{ rad}\right) \approx 2.60\text{ cm}$$
So the length of the minor arc cut off by the chord is approximately 2.60 cm.