MHB Find Side AB of Triangle ABC Given M, N, and C

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To find side AB of triangle ABC given points M, N, and C, the initial assumption of triangle similarity is incorrect, complicating the solution. The sine rule is applied in triangles BNC and BMA, yielding four equations to solve for the unknowns. After eliminating variables, the relation between angles leads to the conclusion that side AB measures 22.5. An alternative approach using coordinate geometry simplifies the problem by establishing a right triangle relationship. The discussion highlights the complexity of the problem while offering different methods to reach the solution.
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                     B
 

                              15

 
A     9     M        11        N   5   C
Triangle ABC, BC = 15, AC = 25.
M and N on AC, such that AM = 9, MN = 11 and CN = 5.
Angle ABM = angle CBN.
Calculate side AB.
 
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It seems as simple as x/9 = 15/5, or am I missing something?
 
pickslides said:
It seems as simple as x/9 = 15/5, or am I missing something?
No; those 2 triangles are not similar...and solution NOT simple: guaranteed!
 
pickslides said:
It seems as simple as x/9 = 15/5, or am I missing something?
Are you claiming that triangles ABM and NBC are similar? Also, it is strange that the fact MN = 11 is not used.
 
View attachment 161

From the sine rule in triangle BNC, $\dfrac{\sin\theta}5 = \dfrac{\sin(\theta+\gamma)}{15}$.

From the sine rule in triangle BMA, $\dfrac{\sin\theta}9 = \dfrac{\sin(\theta+\alpha)}x$.

From the sine rule in triangle ABC, $\dfrac{\sin\alpha}{15} = \dfrac{\sin\gamma}x = \dfrac{\sin(\alpha+\gamma)}{25}$.

That gives four equations for the four unknowns $x$, $\theta$, $\alpha$, $\gamma$. So all you have to do is to solve the equations to find $x$.

In practice, I struggled to do that, but eventually I managed to eliminate $x$ and $\theta$ from the equations and ended with the relation $\sin\alpha = \tfrac23\sin\gamma.$ From there, it was easy to conclude that $\boxed{x = 22.5}.$ (Also, $\cos\alpha = \tfrac{29}{36}$, $\cos\gamma = \tfrac{11}{24}$ and $\tan\theta = \tfrac{\sqrt{455}}{61}.$)
 

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Yep. There is a much easier solution using: A(0,0), C(25,0), B(x,y).
G on AC such that BG perpendicular to AC: AG = x, BG = y (of course).
Let u = angleABM = angleCBN.

TriangleABM: TAN(u) = [y / (x-9) - y/x] / [1 + (y / (x-9))(y/x)]
Do similarly with triangleBCN.
Use above equality and, along with Pythagoras' help with rights ABG and CBG, solve.
 
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