In circle O, chord AB is 9 inches from the center. The diameter of the circle exceeds the length of AB by 2 inches. Find the length of AB

let the chord AB = 2x

then the diameter = 2x+2
or the radius = x+1

This allows me to draw a right-angled triangle with sides x and 9 and hypotenuse x+1
(x+1)^2 = x^2 + 9^2
x^2 + 2x + 1 = x^2 + 81
2x = 80
x = 40

AB = 80

check:
diameter = 82

is 41^2 = 40^2 + 9^2 ?
LS = 1681
RS = 1600+81
= 1681
= LS

YES

To find the length of AB, we can start by considering the properties of a circle and its chords.

1. The distance from the center of the circle to a chord is equal to half the length of the chord. This is known as the "radius-chord theorem."

2. The length of a chord is determined by the perpendicular distance from the center of the circle to the chord. This is known as the "chord theorem."

Let's solve the problem step by step:

Step 1: Label the given information

Let the radius of the circle be r.

Given: the distance from the center of the circle (O) to chord AB is 9 inches.
This implies that the perpendicular distance from O to AB is 9 inches.

Step 2: Use the radius-chord theorem to find the length of AB in terms of r

According to the radius-chord theorem, the distance between the center of the circle and the chord is equal to half the length of the chord. Therefore, we can write:

9 inches = (1/2) * AB

Simplifying the equation, we can find:

2 * 9 inches = AB

AB = 18 inches

Step 3: Use the chord theorem to relate the diameter to the length of AB

According to the chord theorem, the length of a chord is equal to the square root of the product of the diameter and the difference between the diameter and the length of the chord. We can write this as an equation:

AB = √[(diameter) * (diameter - AB)]

Given that the diameter exceeds the length of AB by 2 inches, we can substitute the values:

AB = √[(AB + 2 inches) * (AB + 2 inches - AB)]

AB = √[(AB + 2 inches) * (2 inches)]

Simplify the equation:

18 inches = √[(AB + 2 inches) * 2 inches]

Square both sides:

18^2 inches^2 = [(AB + 2 inches) * 2 inches]

324 inches^2 = [(AB + 2 inches) * 2 inches]

Divide both sides by 2 inches:

162 inches = AB + 2 inches

Subtract 2 inches from both sides:

160 inches = AB

Therefore, the length of AB is 160 inches.

To find the length of AB, we need to set up an equation based on the given information.

Let's define the diameter of the circle as d and the length of AB as x.

From the problem, we know that AB is 9 inches away from the center of the circle. This means that AB is a perpendicular bisector of the diameter.

Since AB is a perpendicular bisector of the diameter, it divides the diameter d into two equal parts. One part is x inches long, and the other part is also x inches long. Therefore, we can say that the diameter d is equal to 2x.

We are also given that the diameter d exceeds the length of AB by 2 inches. This can be written as:
d = x + 2

Now, we can substitute the value of d from the second equation into the first equation to solve for x:
2x = x + 2

Simplifying this equation, we get:
x = 2

Therefore, the length of AB is 2 inches.