In a circle of radius 12cm a chord is drawn 3cm from the centre. Find the angle subtended at the centre
draw the diagram
the distance (3 cm) divided by the radius (12 cm)
... is the cosine of half the subtended angle
Answer
Work
Find the angle subtended at the center of a circle of radius 6.2cm by an arc of length 18.5cm
To find the angle subtended at the center of the circle by the chord, we can use the property that the angle subtended by a chord at the center is twice the angle subtended by the same chord at any point on the circumference.
First, let's find the length of the chord. From the information given, we know that the radius of the circle is 12 cm, and the chord is drawn 3 cm from the center. We can form a right triangle with the radius as the hypotenuse, half of the chord as one side, and the distance from the center to the chord as the other side.
Using the Pythagorean theorem, we have:
(radius)^2 = (distance from center to chord)^2 + (half of chord)^2
(12)^2 = (distance from center to chord)^2 + (3)^2
144 = (distance from center to chord)^2 + 9
144 - 9 = (distance from center to chord)^2
135 = (distance from center to chord)^2
Taking the square root of both sides, we get:
distance from center to chord = √135 ≈ 11.62 cm
Now, we can find the length of the chord:
chord length = 2 × (distance from center to chord)
chord length = 2 × 11.62 cm
chord length ≈ 23.24 cm
To find the angle subtended at the center, we can consider the triangle formed by connecting the center of the circle to the two endpoints of the chord, and calculate the angle using the inverse sine function (sin⁻¹).
angle at center = 2 × sin⁻¹((chord length / 2) / radius)
angle at center = 2 × sin⁻¹(23.24 cm / 24 cm)
Using a calculator, we find:
angle at center ≈ 154.64 degrees
Therefore, the angle subtended at the center of the circle by the chord is approximately 154.64 degrees.