Evil Geometry Problem

Hi, this is Presh Talwalkar. A geometry textbook has the following problem. A right triangle has a hypotenuse equal to 10 and an altitude to the hypotenuse equal to 6. What is the area of the triangle? Can you figure it out? The story goes a teacher asked this problem to a class and every single student quickly submitted a calculation. The teacher then said none of the calculations were correct. Can you figure out why? Give this problem a try and when you're ready keep watching the video for the solution. You can approach this problem by drawing a diagram. We have a right triangle, which has a hypotenuse of 10 and an altitude to the hypotenuse equal to 6. The area of a triangle is equal to its base times its height divided by 2. To apply this formula it might help to rotate the triangle. The hypotenuse of 10 becomes the base and the altitude of 6 becomes the height. Substituting those numbers in the formula we get an area equal to 30. But this is not the correct answer to this problem. The reason is the dimensions of the problem are not correct. Let's see why. Imagine we have a hypotenuse equal to 10. If the two endpoints of this line segment make up two points of the triangle, we want to know where can we draw a third point? Where can the third vertex be to make a right triangle? Now not every point will make a right triangle, but some points will form a right triangle. So we want to figure out the locus of points where we can place the third vertex to make a right triangle. Here's how we can figure it out. If the third vertex forms a 90 degree angle, then it's going to subtend an arc that's a hundred and eighty degrees. So if we think about the hypotenuse as the diameter of a circle, then the third vertex can be anywhere else in the circle to form a right triangle. These are the only points that will form a right triangle with that hypotenuse of 10. The altitude of the hypotenuse is the vertical distance from this third point to the hypotenuse. This will vary depending on which point it is in the circle. So we want to know what's the largest altitude we can make. Well, this will be when we move the altitude all the way to this point. This is the largest possible altitude of the right triangle, and this is exactly equal to the radius of the circle. This is one half the diameter of 10, which means the largest altitude you could have is equal to 5. So the correct answer to this problem is that there's a typo in the textbook. There is no right triangle that has a hypotenuse equal to 10 and an altitude to the hypotenuse equal to 6. At most you can have an altitude equal to half of the hypotenuse, which would be 5. Did you figure it out? Thanks for watching this video. Please subscribe to my channel. I make videos on math and game theory. You can catch me on my blog, Mind Your Decisions, which you can follow on Facebook, Google+, and Patreon. You can catch me on social media at Presh Talwalkar. And if you like this video, please check out my books. There are links in the video description.