This is part of a series on common misconceptions.
True or False?
In a right triangle with hypotenuse 10, the altitude perpendicular to the hypotenuse can be 6.
Why some people say it's true: One of the legs of a right triangle can have a length of 6 (from the Pythagorean triple 6-8-10) and a leg can be the altitude of the triangle.
Why some people say it's false: It's impossible.
The statement is .
According to Thales' theorem, if a triangle is inscribed inside a circle, where one side of the triangle is the diameter of the circle, then the angle opposite to that side is a right angle. The converse of this is also true.
We do know that, in a right triangle, the side opposite to the right angle is the hypotenuse of the right triangle. We can then say that the diameter is the hypotenuse of the right triangle.
In the figure above, there are three right triangles with the same hypotenuse. Consider the hypotenuse with a length of 10. ADB is a right triangle with its other end along with the radius of the semicircle, which is 5. It clearly states that the maximum altitude along the hypotenuse of a right triangle is half the length of the hypotenuse.
Rebuttal: Can't the height be the leg, as in the Pythagorean Triple 6-8-10?
Reply: It can, but considering the altitude perpendicular to the hypotenuse cannot and will never be 6.
Rebuttal: It can be, it's still a triangle.
Reply: Yes, it is still a triangle, but not a right triangle. For example, the triangle --10 is a triangle with an altitude of 6, but it's not a right triangle.