GMAT Math

sanju09 wrote:Triangle ABC has AC = BC, and ∠ACB = 96°. D is a point in triangle ABC such that ∠DAB = 18° and ∠DAB = 30°. What is the measure (in degrees) of ∠ACD?

https://www5.esc13.net/thescoop/math/201 ... can-solve/

That kid certainly does look exceptionally bright. This particular problem is very challenging, certainly beyond the GMAT. Below is an attached diagram.

We know, by the 1800-Triangle Theorem, angle ADB = 132 degrees.

From addition addition, we also know:
angle CAD = 24 degrees
angle CBD = 12 degrees

It appears we have four unknowns, the angles labeled a, b, c, and d. We know
1) at vertex C
a + d = 96
2) In Triangle ACD
24 + a + b = 180
a + b = 156
3) In Triangle BCD
12 + c + d = 180
c + d = 168
4) at vertex D
132 + b + c = 360
b + c = 228

That's four unknowns and four equations, but the trouble is, those four equations are not independent: we get two different ways to get
a + b + c + d = 324

I believe we would need to use trigonometry. We could pick an arbitrary length for AB, and use the Law of Sines to find AC = BC, AD, and BD. From angle CAD, we could use the law of cosines to find length CD, and then use the Law of Sines or Cosines to solve for the angle. That would work, but it's a rather unsatisfying approach, because we would have to use an inverse trig function to give us the angle. There should be a way to do it with pure geometry, without having to use inverse trig functions.

From the software I used to create the diagram, I did find that the angle is 78 degrees. Of course, that's cheating. The computer found the answer, not I.

I would be intrigued to see any purely geometric solution to this problem. Great problem, sanju09!

Mike

Dear dimochka,
I commend you for an elegant trig solution. But since the angle we seek has an integer value, there absolutely must be a clean geometric way to find it directly, without changing it into trig decimals and then changing it back. That's what is truly challenging about this problem.
Mike

Hi All,

Not to put too fine a point on this thread and the news article that serves as the source, but what does this have to do with the GMAT? I'll grant the somewhat interesting aspect of the material (because I'm a nerd), but posting it here only serves to distract Test Takers with beyond-the-boundaries-of-the-GMAT material.

GMAT assassins aren't born, they're made,
Rich