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# Interesting Problem

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Triangle ABC has altitudes AD, BE, and CF. If AD = 12  and BE = 15 is a positive integer, then find the largest possible value of CF?

I have looked at similar problems to this but they are not correct. This needs to be solved relatively quickly if possible.

Thanks,

Guest

May 31, 2023

#1
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The area of triangle ABC is given by

[ABC] = 1/2 * AD * BC = 1/2  BE * AC = 1/2 * CF * AB

Since AD = 12 and BE = 15, we have

[ABC] = 1/2 * 12 * BC = 1/2 * 15 * AC = 1/2 * CF * AB

Dividing these equations, we get

BC/AC = CF/AB

Since BC and AC are positive integers, CF must also be a positive integer. The largest possible value of CF is thus AB, which is equal to

AB = 15 * BC / AC = 15 * 12 / 15 = 12

Therefore, the largest possible value of CF is 12​.

May 31, 2023
#2
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Sadly that answer is incorrect. Nice try though! Guest Jun 1, 2023
#3
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Obtuse scalene triangle.

Sides: a = 362.892   b = 290.314   c = 73.809

Area: T = 2177.353
Perimeter: p = 727.014
Semi-perimeter: s = 363.507

Angle ∠ A = 168.274° = 168°16'27″ = 2.937 rad
Angle ∠ B =9.357° = 9°21'25″ = 0.163 rad
Angle ∠ C =2.369° = 2°22'8″ = 0.041 rad

Height: ha = 12
Height: hb = 15
Height: hc = 59 = altitude CF, which is the maximum positive integer possible.

Jun 1, 2023
#4
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Really good job guest that was correct. Your explanation was amazing

Thanks,

Guest

Guest Jun 1, 2023
#5
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It would be nice to know where the numerical values for a, b and c came from.

Guest Jun 2, 2023
#6
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Suppose that the area of the triangle is A, then (half base times height)

A = AB*CF/2, so AB = 2*A/CF,

A = CA*/BE/2, so CA = 2*A/BE = 2*A/15.

For the triangle to exist,

AB + BC > CA, so 1/CF + 1/12 > 1/15,

BC + CA > AB, so 1/12 + 1/15 > 1/CF,

CA + AB > BC, so 1/15 + 1/CF > 1/12.

The third of these conditions leads to CF < 60 meaning that the largest integer value for CF is 59.

One of the other two conditions gives a minimum for CF.

Jun 2, 2023