SOLUTION: Find the length of the shortest altitude of a triangle with sides of lengths 10, 24, and 26.

In the lesson Proof of the Heron's formula for the area of a triangle in this site you will find how to do it.


Use the Heron's formula.

You have to 1st determine the area of the triangle.

This triangle represents a Pythagorean triple (10-24-26, or 5-12-13), which means that it's a right triangle. No sweat!

Using the 2 legs, you can find the area by applying the formula for the area of a triangle: , which results in: ,

for an area of 120 sq units. Now, if this were NOT a right triangle, the easiest method to find the area is Heron's formula.



Now that we have the area, the shortest altitude, of the three, is the altitude that's drawn to the longest side, 

which in this case is the hypotenuse. After drawing the shortest altitude to the hypotenuse, we now apply the formula 

for the area of a triangle, once again, but this time, the base is the hypotenuse, and the altitude (shortest) is unknown. 

This gives us: 



 

h, or shortest altitude to this triangle = , or