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 \(\text{CE} = 4\).
Show Answer
i. First, we can draw a diagram to help visualize the problem:
As this diagram represents an equilateral triangle, we can use the proper formula to determine the altitude of \(\text{YM}\):
\(h = \cfrac{a\sqrt{3}}{2}\)
\(h = \cfrac{(6)\sqrt{3}}{2}\)
\(h = \cfrac{10.4}{2}\)
\(h = 5.2\)
Therefore, we can determine that the altitude of \(\text{YM}\) is roughly \(5.2\).
ii. First, we can draw a diagram to help visulize the problem:
As this diagram represents an isosceles triangle, we can use the proper formula to determine the altitude of \(\text{YM}\):
\(h = \sqrt{a^2 - \cfrac{b^2}{4}}\)
\(h = \sqrt{8^2 - \cfrac{4^2}{4}}\)
\(h = \sqrt{64 - \cfrac{16}{4}}\)
\(h = \sqrt{64 - 4}\)
\(h = \sqrt{60}\)
\(h = 7.75\)
Therefore, we can determine the altitude of \(\text{TM}\) is roughly \(7.75\).