Explaining the Altitude of a Triangle
The altitude of a triangle is defined as the perpendicular line segment that connects the vertex of the triangle to the side opposite the vertex. Let’s consider the triangle ABC.
Here, the perpendicular line segment, AN is dropped from the vertex A to the side BC. So, AN is the altitude of the triangle from A to BC.
The altitude of a triangle can be one of its sides as well and is observed in the case of a right triangle.
In the right triangle PQR, side PQ is the altitude from P to QR. Also, side RQ is the altitude from R to PQ.
So far we have looked into the altitudes of an acute triangle and a right triangle. Now let’s learn about the altitude of an obtuse triangle.
As seen in the figures, AN is the altitude from A to side BC extended, BM is altitude from B to side AC extended, and CL is the altitude from C to AB.
So the altitudes of an obtuse triangle corresponding to the vertices with acute angles lie outside the triangle. The third altitude from the vertex with an obtuse angle lies inside the triangle.
Altitude of a Triangle in the Area Formula
The formula to calculate the area of a triangle is:
\(Area, A = \frac{1}{2} bh\)
Here, for the base b we use the length of any side of the triangle. The height h will then be the length of the altitude corresponding to that side (base, b) of the triangle.
Solved Examples
Example 1: Find the area of the triangle.
Solution:
Here, AD is the altitude from A to BC.
\(Area, A = \frac{1}{2} bh\) [Formula for area of triangle]
\( = \frac{1}{2}\text{ }18.10\) [Substitue values]
\( = 90\text{ }sq. \text{ }cm\) [Simplify]
So, the area of the triangle is 90 square centimeters.
Example 2: Find the length of the altitude AN of the triangle if its area is 100 square centimeters and side BC measures 12 centimeters.
Solution:
Here, AN is the altitude from A to BC. We can use the formula for the area of a triangle with side BC as base and altitude AN as the height.
\(Area, A = \frac{1}{2}\text{ }bh\) [Formula for area of triangle]
\(100 = \frac{1}{2}\text{ }12.h\) [Substitue value]
\(100 = 6.h\) [Simplify]
\(h = \frac{100}{6}\) [Solve for h]
\(h = 16.66\text{ }cm\) [Solve for h]
So, the altitude AN of the triangle measures 16.66 centimeters.
A triangle can have three altitudes, one from each vertex to the side opposite the vertex.
It is important to mention the vertex and the side when defining the altitude of a triangle as altitudes from different vertices and sides can have different measures.
The point at which all three altitudes of a triangle intersect is called the orthocenter of the triangle.
The altitude of an obtuse triangle lies outside the triangle. So the given statement is false.