1. What is an Altitude?
An altitude of a triangle is a line segment drawn from any vertex perpendicular to the opposite side (or to the line containing the opposite side). This means the altitude forms a 90° angle with the base.
A
/|
/ | h (altitude)
/ |
B---C
The length of the altitude is often used to calculate the area of a triangle.
2. Definition of Altitude
Definition: An altitude of a triangle is a perpendicular drawn from a vertex to the opposite side (called the base) or to the line containing that side.
In \(\triangle ABC\):
- Altitude from vertex \(A\) meets \(BC\) at a right angle.
- Altitude from vertex \(B\) meets \(AC\) at a right angle.
- Altitude from vertex \(C\) meets \(AB\) at a right angle.
3. Altitudes in Different Types of Triangles
The nature of the altitude changes depending on the type of triangle:
3.1. In Acute-Angled Triangles
All three altitudes lie inside the triangle because all angles are less than \(90^\circ\).
3.2. In Right-Angled Triangles
If the right angle is at \(B\), then the two legs \(AB\) and \(BC\) themselves act as altitudes.
The third altitude (from the right-angle vertex to the hypotenuse) lies inside the triangle.
3.3. In Obtuse-Angled Triangles
Because one angle is greater than \(90^\circ\), the altitude from that obtuse vertex falls outside the triangle when extended to meet the base line.
4. Point of Concurrency: Orthocentre
The three altitudes of a triangle always intersect at a single point called the orthocentre.
A
/|\
/ | \
/ | \
B---O---C (O = orthocentre)The location of the orthocentre changes based on the type of triangle:
- Inside an acute triangle
- At the right-angle vertex in a right triangle
- Outside an obtuse triangle
5. Why Altitudes Are Useful
Altitudes help in:
- Calculating the area of a triangle using: \(\dfrac{1}{2} \times \, \text{base} \times \, \text{height}\)
- Constructing geometric shapes
- Understanding triangle properties
- Locating the orthocentre
They also play an important role in proofs and geometric constructions.