A chord is drawn 3cm away from the centre of a circle of radius 5cm .calculate the length of the chord
To calculate the length of the chord, we can use the Pythagorean theorem.
Let's label the center of the circle as point O, and the endpoints of the chord as points A and B.
Since the chord is drawn 3cm away from the center of the circle, we can draw a perpendicular line from the center to the chord. Let's label the point where this perpendicular line intersects the chord as C.
We can form a right triangle by connecting points O, C, and A.
The length of OA (the radius) is 5cm, and the length of OC (the perpendicular line) is 3cm.
Using the Pythagorean theorem:
AC^2 + OA^2 = OC^2
AC^2 + 5^2 = 3^2
AC^2 + 25 = 9
AC^2 = 9 - 25
AC^2 = -16
Since the length of a line segment cannot be negative, we have made an error in our calculations.
Let's try a different approach.
Instead of connecting points O, C, and A, we can draw a line segment from point O to point B.
This creates a right triangle with length OC as the base and length OB as the hypotenuse.
Using the Pythagorean theorem:
OC^2 + OB^2 = BC^2
3^2 + 5^2 = BC^2
9 + 25 = BC^2
34 = BC^2
Taking the square root of both sides, we find:
sqrt(34) = BC
Therefore, the length of the chord is approximately 5.83 cm.
To calculate the length of the chord, we can use the formula for the length of a chord in a circle.
The formula is given by:
Length of chord = 2 * radius * sin(angle/2)
In this case, the radius of the circle is 5cm. We need to find the angle corresponding to the chord that is 3cm away from the center.
To find the angle, we can use the inverse sine function:
angle = 2 * sin^(-1)(chord length / (2 * radius))
Plugging in the values, we have:
angle = 2 * sin^(-1)(3 / (2 * 5))
= 2 * sin^(-1)(0.3)
≈ 2 * 17.46° (rounded to two decimal places)
Now we can substitute the angle value into the original formula to calculate the length of the chord:
Length of chord = 2 * radius * sin(angle/2)
= 2 * 5 * sin(17.46° / 2)
≈ 2 * 5 * sin(8.73°)
≈ 2 * 5 * 0.1504 (rounded to four decimal places)
≈ 1.504 cm (rounded to three decimal places)
Therefore, the length of the chord is approximately 1.504 cm.
To calculate the length of a chord in a circle, you can use the following formula:
Length of chord = 2 * Radius * sin(angle/2)
In this case, the radius of the circle is given as 5cm, and the chord is drawn 3cm away from the center. To find the angle, we can use the fact that the chord is 3cm away from the center, which means it is also 3cm away from the radius. This forms a right triangle with one side of length 3cm and the hypotenuse of length 5cm.
Using the Pythagorean theorem, we can find the length of the other side of the right triangle:
3^2 + x^2 = 5^2
9 + x^2 = 25
x^2 = 25 - 9
x^2 = 16
x = √16
x = 4
So, the length of the other side of the right triangle is 4cm. Now we can find the angle:
sin(angle) = opposite/hypotenuse
sin(angle) = 4cm/5cm
sin(angle) = 0.8
To find the angle itself, we can use the inverse sine function (sin^-1) on a calculator:
angle = sin^-1(0.8)
angle ≈ 53.13 degrees
Now we can substitute these values into the formula to find the length of the chord:
Length of chord = 2 * 5cm * sin(53.13/2)
Length of chord = 10cm * sin(26.57)
Length of chord ≈ 10cm * 0.447
Length of chord ≈ 4.47cm
Therefore, the length of the chord is approximately 4.47cm.