In a circle of radius 6cm, a chord 3cm is drawn 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 by the chord
Explain how
Draw a diagram and you can see that
The chord subtends an angle of 2θ, where
cosθ = 3/6
and the arc length is s = 2rθ
Matthew
Has
Yes
(a) Oh, it's always a party when chords start getting drawn in circles! So, let's get this circus started. The radius of the circle is 6cm, and the chord is 3cm. To calculate the angle subtended by the chord at the center, we need to bring in the comedy equation known as "angle subtended by a chord." This equation states that the angle subtended by a chord is double the angle subtended by the same chord at any point on the circumference. In this case, since the chord is drawn from the center, the angle subtended by the chord at the center is double the angle subtended by the chord on the circumference. Now, let's put on our clown noses and calculate the angle!
The angle subtended by the chord at any point on the circumference is given by:
angle = 2 * arcsin(chord / (2 * radius))
Plugging in the values, we have:
angle = 2 * arcsin(3 / (2 * 6))
Simplifying:
angle = 2 * arcsin(1 / 4) [arcsin(1/4) is approximately 14.48 degrees]
angle ≈ 2 * 14.48
angle ≈ 28.96 degrees
So, the angle subtended by the chord at the center of the circle is approximately 28.96 degrees.
(b) Now that we have the angle subtended by the chord at the center, we can calculate the length of the minor arc cut off by the chord. The length of a minor arc is given by the formula:
arc length = (angle / 360) * 2 * pi * radius
Plugging in the values:
arc length = (28.96 / 360) * 2 * pi * 6
arc length ≈ (0.080°) * 2 * pi * 6 [approximately 0.080° is the ratio of 28.96 to 360]
arc length ≈ 0.080 * 2 * 3.14 * 6
arc length ≈ 0.080 * 37.68
arc length ≈ 3.0144 cm
So, the length of the minor arc cut off by the chord is approximately 3.0144 cm. Join the circus, and enjoy the show!
To solve this problem, we can use the properties of circles and triangles. Let's break it down step by step:
a) To calculate the angle subtended by the chord at the center of the circle, we can use the fact that an inscribed angle is half the measure of the arc it intercepts.
In this case, the chord divides the circle into two arcs: a minor arc and a major arc. Since the chord is drawn from the center, it bisects the minor arc.
To find the angle subtended by the chord at the center, we need to find the measure of the minor arc. We can do this by using the formula for the length of an arc:
Length of arc = (angle/360) * 2πr
Since the radius of the circle is 6cm, the length of the minor arc will be:
Length of minor arc = (angle/360) * 2π(6)
To find the angle, we can rearrange the formula:
angle = (Length of minor arc / (2πr)) * 360
Plugging in the values, we have:
angle = (Length of minor arc / (2π*6)) * 360
b) The length of the minor arc cut off by the chord is the length of the chord itself. In this case, the chord has a length of 3cm.
Therefore, the angle subtended by the chord at the center of the circle is given by:
angle = (3 / (2π*6)) * 360
Now, we can calculate the values:
angle = (3 / (12π)) * 360
Using a calculator:
angle ≈ 46.84 degrees
(b) The length of the minor arc cut off by the chord is simply the length of the chord itself, which we already know to be 3cm.