The diameter of a circle is 130 cm, and a chord of the circle is 66 cm long. What is the distance between the chord and the center of the cirlce?
Diameter = 130 cm
Radius = 130/(2) = 65 cm
Chord = 66 cm
half chord = 33 cm
The centre forms an isosceles triangle with the chord, with the two equal sides the radii of the circle.
Consider the right triangle which forms half of the isosceles triangle.
distance from centre to the chord = h
where h²+33²=65²
Solve for h. (Hint: h is an integer)
To find the distance between the chord and the center of the circle, we first need to determine the radius of the circle.
The diameter of a circle is equal to twice the radius. In this case, the diameter is given as 130 cm. Therefore, 130 cm = 2 * radius. To isolate the radius, we divide both sides of the equation by 2: radius = 130 cm / 2 = 65 cm.
Now, we can draw a diagram to visualize the problem. The chord divides the circle into two segments. The distance between the chord and the center of the circle is equal to the perpendicular distance from the center to the chord.
Using the property of circles, we know that the perpendicular drawn from the center of a circle to a chord bisects the chord. Therefore, the chord is divided into two equal segments of length 66 cm each.
To find the distance between the chord and the center of the circle, we can consider one of the right-angled triangles formed by this line, the radius, and half of the chord.
In the triangle, the length of the hypotenuse is the radius (65 cm), one side is half the length of the chord (66 cm / 2 = 33 cm), and the other side is the distance we want to find.
We can use the Pythagorean theorem to find this distance. The theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Applying the theorem, we have: distance^2 + (33 cm)^2 = (65 cm)^2.
To solve for the distance, we subtract (33 cm)^2 from both sides of the equation: distance^2 = (65 cm)^2 - (33 cm)^2.
Calculating this expression, we get: distance^2 = 4225 cm^2 - 1089 cm^2 = 3136 cm^2.
Finally, we take the square root of both sides of the equation to find the distance: distance = √(3136 cm^2) = 56 cm.
Therefore, the distance between the chord and the center of the circle is 56 cm.