In a circle of radius 6cm a chord is drawn 3cm from the centre.
A. Calculate the angle substended by the chord at the centre of the circle.
B. Hence find the length of the minor arc cut off by the chord.
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
oobleck
May 3, 2021
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To solve this problem, we can use the properties of a circle and apply geometry concepts. Let's break it down step by step:
A. To calculate the angle subtended by the chord at the center of the circle, we need to use the relationship between the central angle formed by the chord and the radius of the circle.
1. First, we need to find the length of the chord. We are given that the chord is drawn 3 cm from the center, and the radius of the circle is 6 cm. Since the chord is parallel to the circumference of the circle, it splits the radius into two equal parts.
Using the Pythagorean theorem, we can calculate the length of the chord:
Length of the chord = 2 × √(radius² - distance from center²)
= 2 × √(6² - 3²)
= 2 × √(36 - 9)
= 2 × √27
= 2 × 3√3
= 6√3 cm
2. Next, we can use the relationship between the chord and the central angle to calculate the angle subtended at the center.
The angle subtended by a chord at the center is twice the angle subtended at any point on the circumference.
Using this information, we can set up the following equation:
Length of the chord / Radius of the circle = 2 × Angle subtended by the chord at the center
Plugging in the values, we have:
(6√3 cm) / (6 cm) = 2 × Angle
√3 / 1 = 2 × Angle
Now, solve for Angle:
Angle = (√3 / 2) radian
Hence, the angle subtended by the chord at the center is (√3 / 2) radian.
B. To find the length of the minor arc cut off by the chord, we can use the formula:
Length of the minor arc = Radius × Angle subtended by the chord at the center
Plugging in the values, we have:
Length of the minor arc = 6 cm × (√3 / 2) radian
= 3√3 cm
Therefore, the length of the minor arc cut off by the chord is 3√3 cm.