In a circle of radius 6cm,a chord is drawn 3cm from the centre
Find the length of the minor arc cut off the chord.
We can start by drawing a diagram:
In the diagram, the chord is labelled as AB and the distance from the centre to the chord is labelled as CD. We want to find the length of the minor arc cut off by chord AB.
Since CD is perpendicular to AB, we can use the Pythagorean theorem to find its length.
CD^2 + BC^2 = BD^2
CD^2 + (6 - 3)^2 = 6^2
CD^2 + 9 = 36
CD^2 = 27
CD = sqrt(27) = 3sqrt(3)
Now, we can use the fact that the angle subtended by a chord at the centre of a circle is double the angle subtended by the same chord at any point on the circumference of the circle.
Let E be the midpoint of AB. Then angle CEB is half the angle subtended by chord AB at the centre of the circle.
Using trigonometry, we can find the value of sin(CEB):
sin(CEB) = CD / CE
sin(CEB) = 3sqrt(3) / 6
sin(CEB) = sqrt(3) / 2
We know that sin(60 degrees) = sqrt(3) / 2, so we can conclude that angle CEB = 60 degrees.
Therefore, the angle subtended by chord AB at the centre of the circle is 2 x 60 = 120 degrees.
Since the circumference of a circle is 2π times its radius, the circumference of this circle is 2π x 6 = 12π.
So the fraction of the circumference that is cut off by minor arc AB is:
120 / 360 = 1/3
Therefore, the length of the minor arc cut off by chord AB is:
1/3 x 12π = 4π
So the length of the minor arc cut off by chord AB is 4π cm.