in the diagram below of Circle of chord AB bisects chord CD at E. If AE equals 8 and BE equals 9 find the length of CD in simplest radical form
http://www.mathopenref.com/chordsintersecting.html
AB*BE=CE*DE
72=CE*DE but CE=DE=CD/2 (it was bisected)
72=(CD/2)^2
CD=you can do it from here.
To find the length of CD, we can use the intersecting chords theorem, which states that when two chords intersect in a circle, the products of their segments are equal.
Given that AE equals 8 and BE equals 9, we can set up the following equation using the intersecting chords theorem:
AE * BE = CE * DE
Substituting the given values:
8 * 9 = CE * DE
Simplifying:
72 = CE * DE
Since chord AB bisects chord CD at E, CE is equal to DE. Let's use variable x to represent the length of CE (or DE):
72 = x * x
Solving for x, we get:
x^2 = 72
Taking the square root of both sides:
x = √72
Since we want the length of CD, which is equal to 2 times x (since E is the midpoint of CD), we can substitute back in:
CD = 2 * √72
Simplifying:
CD = 2 * √(9 * 8)
CD = 2 * √9 * √8
CD = 2 * 3 * √8
Finally, simplifying the expression:
CD = 6√8
So, the length of CD in simplest radical form is 6√8.
To find the length of CD in simplest radical form, we can use the property that when two chords intersect inside the circle, the product of their respective segments is equal.
In this case, we have AE = 8 and BE = 9. Let's assume the length of AE is x and the length of CE is y. So, the length of BE would be (8 + y) since AE bisects CD.
Using the property mentioned earlier, we can set up the equation:
AE * BE = CE * DE
8 * (8 + y) = 9 * y
Now, let's solve the equation for y:
64 + 8y = 9y
64 = y
So, the length of CE is 64.
But we need to find CD, which is the sum of CE and DE. Since AE bisects the chord, we can assume that DE is also 64.
Therefore, the length of CD is 64 + 64 = 128.