# In a right triangle with hypotenuse 10, can the altitude perpendicular to the hypotenuse be 6?

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 $\color{#D61F06}{\textbf{false}}$.

Proof: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.

$_\square$

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 hypotenusecannotandwill neverbe 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 $\small\sqrt{61}$-$\small\sqrt{61}$-10 is a triangle with an altitude of 6, but it's not a right triangle.

**See Also**

**Cite as:**In a right triangle with hypotenuse 10, can the altitude perpendicular to the hypotenuse be 6?.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/can-a-right-triangle-with-hypotenuse-10-have-a/