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# Trig challenge problem: area of a triangle

Sal solves a very complicated geometrical trig problem that appeared as problem 11 in the 2003 AIME II exam. Created by Sal Khan.

## Want to join the conversation?

• The confusing part for me is that the question says point D is on the same side of AB as C. Is that badly worded or is it just me?
• It's not just you. That is where I started to get lost, too. I watched the video a second time and suddenly it all made sense! They were just trying to tell us that the point D is inside the triangle ABC.
• At , how did you get 2 (25/2) ^2 (1 + sine theta) from 2 (25/2) ^2 + 2 (25/2)^2 sine theta?
• The expression 2(25/2)^2 + 2(25/2)^2sin theta has two terms. Both of these terms has a common factor 2(25/2)^2. Sal "pulled" this factor out of both terms.

Let's make a substitution so it might be easier to see: let x = 2(25/2)^2. Then the expression becomes x + xsin theta. We pull the x out to become x(1 + sin theta). This is simple factoring out, or Sal sometimes calls it undistributing. We can redistribute the x to both terms by multiplying through to get x + xsin theta.
• The Point "D" should not be positioned inside the triangle. 15+15 = 30 which is only 1 less than 24+7 which makes you wonder how it could be inside the triangle. Note that 5*11^1/2/2 is greater than 8. Compare with the triangle base of 7. Did I get something wrong here?
• I was just wondering this exact same thing. It's physically impossible to have that triangle inside the given triangle. I even took it to graph paper and used a compass and it's impossible to have point D where he drew it.
• At , how can he deduce that it will be a right triangle and that line DM is perpendicular to line AB?
• He knows it is a right triangle, because if you draw a line from the `midpoint` of an isosceles triangle to its apex, it is by definition a right triangle. That midpoint means that you form two triangles such that the angles on either side of that midpoint add up to 180 degrees, plus all their corresponding sides and angles are the same. Two equal angles that add up to 180 degrees have to be 90 degrees. Ta da!
Draw a couple of isosceles triangles (two identical sides) and you can remind yourself that this is true. Some of Sal's great videos in the geometry topics can review proofs of this for you. It makes life easier when working with triangles to be confident of that fact.
• I've been in similar situations so here are some things I'd like to share (hopefully this isn't too much for a reply to an "off-topic" question) :
1) Trigonometry is like learning the alphabet compared to the famous problems that Euler, Ramanujan and Gauss solved. Some kids learn the alphabet faster and know more words, but that doesn't determine if they would go on to become great novelists.
2) At this stage it's more about practice and less about inherent skill. Many university students with iqs of about 120 would be able to solve this problem in 4 mins, simply because they have practiced more. It's too early to be equating mathematical success with your genius.
3) I think if someone only knew the basics of trig (sin,cos,tan) and tried to attempt this problem, they would take much time to answer it because the number of ways to think about approaching this problem with only that information is huge. Imagine Ramanujan with the mathematical information and exposure of the average 15 year old but leave himself with his own IQ. I think he would take way more than 4 mins.
• Right at the end, when asked to add m+n+p, I would have used sqrt11 to represent n, as it was derived as such, not just n: why did Sal use just 11 rather than Sqrt11?
• it says in the intro that n has to be an integer, n's root would not be an integer.
• , I begin to loose out, what is that law of cosines that sal is referring to?
• At , its cos(x)=sin(90-(x)) cos(theta+90), so why does it become sin(90-(theta-90))? Shouldnt it be sin(90-(theta PLUS 90)) instead?
• If you look at it this way, it should help: cos(theta+90)=sin(90-(theta+90)). Which is the same as sin(90-theta-90).
• I haven't finished the last unit but I have no idea how to solve this. After finishing the last unit would i have an idea of how to solve this?