Mind your decisions, I'm Presh Talwalkar.
In square ABCD,
draw lines BE, BF and FE as shown. If the areas of the triangles are 3,4, and 5,
what is the area of triangle BEF?
This is adapted from a Tweet by the National Museum of Mathematics.
For a little bit more of a challenge, consider the general case where ABCD is a rectangle,
and the areas are x, y, and z.
What is the area of BEF in terms of x, y, and z? This comes from a Tweet by Mustafa Kemal Ince.
Pause the video if you'd like to give this problem a try, and when you're ready, keep watching to learn how to solve this problem.
We'll go ahead, and tackle the general case, and we'll prove the following formula.
To do that, we'll work step by step.
First, consider this triangle.
Suppose one side of the rectangle is a, and the other side of the triangle is c.
Because the triangle has an area of y, ac/2 = y. This means c = 2y/a.
Now, suppose the other side of the rectangle is b. The remaining distance on this side will be b - (2y/a).
Now consider the triangle, which has an area of x.
If its other side is equal to d, we can solve that bd/2 = x.
This means d = 2x/b.
The other distance on this side will be a - 2x/b.
We have the final triangle, with an area of z.
So we can set that 1/2 * each of its legs = the area of z.
We'll then focus on this equation.
First, we multiply both sides by 2, and then we'll distribute this binomial.
Now it looks pretty complicated, but it's going to simplify.
The first thing we'll do is we'll subtract 2z from both sides. We'll then factor 2 from these 3 terms.
Now, let's multiply both sides by the quantity ab.
On the right-hand side, this will vanish, because 0 * ab = 0.
On the left-hand side, we'll distribute it over 3 different terms.
The first term becomes ab(the quantity)^2.
In the second term, we add a factor of ab, and in the third term, the denominator of ab will vanish.
This equation looks complicated, but it's actually something you should be able to solve.
This is going to be a quadratic equation in the variable ab.
So, we can use the quadratic formula to solve it.
ab = (the opposite of the coefficient on ab
± √4(x+y+z)^2-16xy)/2.
We then will simplify the square root, by factoring out a 2, and then these factors of 2 will all cancel out.
So ab = (x + y + z) ± the following square root.
Now it looks like we have two choices; plus or minus. But, here's how we can rule out one of the choices.
Recall that ab = the area of the rectangle, and that has to be larger than x + y + z,
because those are the areas of those three triangles.
Since ab > (x+y+z), we will need the positive square root
(we don't want to subtract anything from it, the area is going to be larger than the areas of the 3 triangles)
So, we eliminate the negative term, and we have an answer for what ab is equal to.
Now, go back to the original diagram.
The entire area of the rectangle is ab.
So ab will be equal to the entire area,
but it's also going to be equal to the sum of the areas of these triangles.
So it's also equal to (x+y+z+w).
We have two different equations for ab;
therefore, we must have the following 2 terms be equal.
This means w = √(x+y+z)^2 - 4xy.
That's going to be the formula we use.
Now, go back to the original problem, where we have 3, 4, and 5.
We can easily solve for w, by substituting in and then simplifying: w = 4 * √6, which approximates to 9.80,
, and that's our answer.
I think it's incredible that there's such a simple formula to solve all problems of this type!
Thanks for making Mind your Decisions one of the best channels on YouTube. As always, thanks for watching, and thanks for your support.