Two chords intersect a circle. The shorter chord is divided into segments of lengths of 9 inches and 12 inches. The longer chord has a length of 24 inches. Find the length of the shorter portion of the longer chord .
make a sketch
angles subtended by the same chord are equal,
so you have two similar triangles
let the 24 inch chord be divided into x and 24-x
then by ratios ...
x/12 = 9/24-x
108 = 24x - x^2
x^2 - 24x + 108 = 0
(x-6)(x-18)=0
x = 6 or 18
if one piece is 6, the other is 24-6 = 18
the shorter piece is 6
To solve this problem, we can use the property of intersecting chords in a circle. According to this property, when two chords intersect in a circle, the product of the lengths of the segments on one chord is equal to the product of the lengths of the segments on the other chord.
Let's denote the length of the shorter portion of the longer chord as 'x'. We can set up the following equation based on the property mentioned above:
9 * 12 = x * (24 - x)
Simplifying this equation, we have:
108 = x * (24 - x)
Expanding the right side of the equation, we get:
108 = 24x - x^2
Rearranging the equation, we have:
x^2 - 24x + 108 = 0
Now, we can solve this quadratic equation using factoring, completing the square, or using the quadratic formula. In this case, let's use factoring:
(x - 6)(x - 18) = 0
Setting each factor equal to zero, we have:
x - 6 = 0 --> x = 6
x - 18 = 0 --> x = 18
Since we are looking for the shorter portion of the longer chord, the length cannot be 18 inches (as it would make the portion longer than the entire chord). Therefore, the length of the shorter portion of the longer chord is 6 inches.