# Altitude

The altitude is better known as the height of a triangle. It is a perpendicular line segment that directly connects a vertex to a point on the opposite side by creating a right/90° angle with that point. It can be represented visually as such:

## Altitude Formulas

Finding the altitude of a triangle depends on the type of triangle we're investigating. The different ways of finding the altitude can be determined by the following formulas. Here is a refresher for the different types of triangles we are investigating:

Shape Formula Notes
Scalene Triangle $$h = \cfrac{2\sqrt{s(s - a)(s - b)(s -c)}}{b}$$ $$a$$, $$b$$ and $$c$$ represent the side lengths of the triangle
Iscoceles Triangle $$h = \sqrt{a^2 - \cfrac{b^2}{4}}$$ $$b$$ represents the base of the triangle. $$a$$ represents one of the other 2 sides
Equilateral Triangle $$h = \cfrac{a\sqrt{3}}{2}$$ $$a$$ represents one of the side lengths
Right Triangle $$h = \sqrt{xy}$$ $$x$$ and $$y$$ represent the individual bases of the triangle

Example

Determine the altitude of the following triangle from Point $$\text{E}$$:

We can determine that the triangle is scalene as none of its side lengths are the same. In order to determine the value of $$s$$, we need to use the following formula:

$$s = \cfrac{a + b + c}{2}$$

$$s = \cfrac{7 + 15 + 12}{2}$$

$$s = \cfrac{34}{2}$$

$$s = 17$$

As we have determined that $$s = 17$$, we can use the proper formula to determine the altitude of $$\text{AM}$$:

$$h = \cfrac{2\sqrt{s(s - a)(s - b)(s -c)}}{b}$$

$$h = \cfrac{2\sqrt{17(17 - 7)(17 - 15)(17 - 12)}}{15}$$

$$h = \cfrac{2\sqrt{17(10)(2)(5)}}{15}$$

$$h = \cfrac{2\sqrt{1700}}{15}$$

$$h = \cfrac{82.462}{15}$$

$$h = 5.497$$

Therefore, we can determine that the altitude of the triangle is roughly $$5.497$$.

Determine the altitudes of the following triangles:
i. In △$$\text{OPQ}$$, the altitude of $$\text{YM}$$ where $$\text{XY} = 6$$, $$\text{YZ} = 6$$ and $$\text{XZ} = 6$$.
ii. In △$$\text{CET}$$, the altitude of $$\text{TM}$$ where $$\text{CT} = 8$$, $$\text{ET} = 8$$, and $$\text{CE} = 4$$.