Let AB be a diameter of a circle, and let C be a point on the circle such that AC = 8 and BC = 4. The angle bisector of angle ACB intersects the circle at point M. Find CM.
The diagram:
https://artofproblemsolving.com/texer
The previous posts on this topic were incorrect or had the wrong values. I was also unable to find out the values from their work.
Thanks,
Guest
To find CM, we can use the angle bisector theorem. According to the angle bisector theorem, the ratio of the lengths of the segments formed by the angle bisector on a triangle is equal to the ratio of the lengths of the opposite sides.
In triangle ABC, let CM intersect AB at point D. We know that AD + DB = AB (the full length of the diameter). Since AB is the diameter, it is twice the radius of the circle.
Let's assign the radius as r. Therefore, AB = 2r.
Given AC = 8 and BC = 4, we can determine AD and DB.
AD/DB = AC/BC AD/DB = 8/4 AD/DB = 2
Since AD + DB = AB = 2r, we have:
AD = 2/(2 + 1) * AB AD = 2/3 * 2r AD = 4r/3
Similarly,
DB = 1/(2 + 1) * AB DB = 1/3 * 2r DB = 2r/3
Now, we can find the length of CM.
CM = CD + DM
Since CM is the angle bisector, we have:
CD/DB = CM/MB
CD/DB = CM/(AB - CM)
Substituting the values we have:
4r/3 / (2r/3) = CM / (2r - CM)
4r/3 * (3/2r) = CM / (2r - CM)
2 = CM / (2r - CM)
2(2r - CM) = CM
4r - 2CM = CM
4r = 3CM
CM = 4r/3
Now we need to find the value of r. Since AC = 8 and BC = 4, we can use the Pythagorean theorem on triangle ABC:
AC^2 + BC^2 = AB^2
8^2 + 4^2 = (2r)^2
64 + 16 = 4r^2
80 = 4r^2
r^2 = 80/4
r^2 = 20
r = √20
Now we can find CM:
CM = 4r/3
CM = 4(√20)/3
Simplifying the expression:
CM = (4/3) * (2√5)
CM = (8√5)/3
Therefore, CM is equal to (8√5)/3 units.
