I need help with altitudes
Triangle ABC has altitudes AD, BE, and CF. If AD = 12, BE = 12, and CF is a positive integer, then find the largest possible value of CF.
To find the largest possible value of CF, we can use the property that the product of two segments of intersecting altitudes is constant.
Let's denote the length of CF as x.
According to the property, we have:
AD * BD = CD * FD
Given that AD = 12 and BD = 12, we can substitute these values into the equation:
12 * 12 = CD * FD
144 = CD * FD
To find the largest possible value of CF (which is CD), we should make FD as small as possible. Since FD is also an altitude, it should be a positive integer. The smallest positive integer value for FD is 1.
So, when FD = 1:
144 = CD * 1
CD = 144
Therefore, the largest possible value of CF (CD) is 144.
