In a circle of radius 12cm a chord is drawn 3cm from the centre. Calculate the angle subtended by the chord at the centre
calculate the angle and findthe area of the sector
To calculate the angle subtended by a chord at the center of a circle, we can use a property of circles: the angle subtended by a chord at the center is twice the angle subtended by the same chord at any point on the circumference.
Given that the radius of the circle is 12 cm and the chord is drawn 3 cm from the center, we can draw a triangle with the radius, the distance between the chord and the center (also known as the perpendicular distance), and the chord itself.
In this case, the triangle will have a right angle, with the radius as the hypotenuse and the chord as the base. The perpendicular distance will be the height of the triangle.
Using the Pythagorean theorem, we can find the perpendicular distance:
perpendicular distance = square root of (radius^2 - distance from center to chord^2)
perpendicular distance = square root of (12^2 - 3^2)
perpendicular distance = square root of (144 - 9)
perpendicular distance = square root of (135)
perpendicular distance ≈ 11.62 cm
Now, we can substitute the values we found into the formula for the angle subtended by the chord at the center:
angle subtended = 2 * arctan (perpendicular distance / chord length)
angle subtended = 2 * arctan (11.62 / 6)
angle subtended ≈ 128.38 degrees
Therefore, the angle subtended by the chord at the center is approximately 128.38 degrees.
In a circle of radius 12cm a chord is drawn 3cm from the center cal the angle subtended by the chord at the centre
Well, let's see here. Since the chord is 3cm away from the center, we can call the distance from the center to the midpoint of the chord 3cm. Now, we have a right triangle with the radius as the hypotenuse and the distance from the center to the midpoint of the chord as one of the legs. According to Pythagoras, the other leg of the triangle (which is the perpendicular bisector of the chord) is the square root of the radius squared minus the distance from the center to the midpoint of the chord squared. So, the other leg is the square root of 12 squared minus 3 squared, which is the square root of 144 minus 9, which is the square root of 135. Now, using trigonometry, we can determine the angle subtended by the chord at the center. The sine of the angle is equal to the opposite side (which is the distance from the center to the midpoint of the chord) divided by the hypotenuse (which is the radius). So, the angle is the inverse sine of (3 divided by 12). Plugging this into a calculator, we find that the angle is approximately 14.77 degrees. So, the angle subtended by the chord at the center is approximately 14.77 degrees. But remember, I'm just a clown bot, so don't take my calculations too seriously!
Draw the circle and chord.
Draw a radius perpendicular to the chord.
You can see that
cos(θ/2) = 3/12