A diameter of a circle is 32 yards long. A chord is 11 yards from the center. How long is the chord?
diameter = 32
r = 16
Let the length of the chord be 2x.
Join one end of the chord to the centre.
You now a right-angled triangle with hypotenuse of r, one leg as 11 and the other leg as x
x^2 + 11^2 = (16)^2
x^2 = 135
x = √135
so the chord has length 2√135 or 23.24 yards
To find the length of the chord, we can use the Pythagorean theorem.
The Pythagorean theorem states that in a right triangle, the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse.
In this case, the diameter of the circle is the hypotenuse, and the chord with 11 yards from the center forms a right triangle.
Let's denote the length of the chord as "c", and the distance from the center to the chord as "d". In this case, "d" is equal to half the length of the chord. Therefore, "d" = 11 yards/2 = 5.5 yards.
We can then form a right triangle with the hypotenuse equal to the diameter of the circle (32 yards), one leg equal to the distance from the center to the chord (5.5 yards), and the other leg equal to half the length of the chord (c/2).
Using the Pythagorean theorem:
(32 yards)^2 = (5.5 yards)^2 + (c/2)^2
1024 yards^2 = 30.25 yards^2 + (c/2)^2
Subtracting 30.25 yards^2 from both sides:
993.75 yards^2 = (c/2)^2
Taking the square root of both sides:
sqrt(993.75) yards = c/2
Multiplying both sides by 2:
2 * sqrt(993.75) yards = c
So, the length of the chord is approximately 62.91 yards.