Altitude of a Triangle

The altitude of a triangle is a perpendicular segment drawn from a vertex to the line containing the opposite side (base). It represents the height of the triangle relative to that base.

1.0Altitude of a Triangle Definition

The Altitude of a Triangle is a perpendicular segment or line drawn from a vertex of the triangle to the line that extends through the opposite side (known as the base). This segment represents the height of the triangle with respect to that base. Each triangle has three altitudes, one from each vertex, which may lie inside or outside the triangle depending on the type of triangle.

2.0Properties of the Altitude of a Triangle

1. Perpendicularity: Each altitude is perpendicular to the side of the triangle it meets. This means it forms a right angle (90 degrees) with the side.
2. Intersection Point: The three altitudes of a triangle intersect at a single point called the orthocenter. The orthocenter can be inside, outside, or on the triangle depending on the type of triangle (acute, obtuse, or right, respectively).
3. Number of Altitudes: Every triangle has exactly three altitudes, one from each vertex.
4. Relation to Area: The altitude is used in the formula to calculate the area of the triangle:

$\text{ Area }=\frac{1}{2}\times \text{ base }\times \text{ height }$

5. Altitude in Different Triangles:

• Acute Triangle: All altitudes lie inside the triangle.
• Right Triangle: Two altitudes are the legs of the triangle, and the third altitude (from the right-angle vertex to the hypotenuse) is inside the triangle.
• Obtuse Triangle: Two altitudes lie outside the triangle, and one lies inside.

6. Equilateral Triangle: In an equilateral triangle, all 3 altitudes are equal in length and also serve as medians and angle bisectors.
7. Isosceles Triangle: In an isosceles triangle, the altitude from the vertex angle (the angle between the equal sides) also bisects the base and the vertex angle.
8. Altitude and Orthocenter Relationship: The altitudes of a triangle are concurrent, meaning they intersect at the orthocenter, which has different positions based on the type of triangle (inside for acute, on the hypotenuse for right, and outside for obtuse triangles).

3.0Altitude of Types of Triangles

Let us discuss some types of Triangles

1. Altitude of Scalene Triangle
2. Altitude of Isosceles Triangle 
3. Altitude of an Equilateral Triangle
4. Altitude of a Right Triangle
5. Altitude of an Obtuse Triangle

Altitude of a Scalene Triangle

In a Scalene Triangle, where all sides and angles are unequal, the altitude is a perpendicular segment drawn from a vertex to the line containing the opposite side. Each vertex has a corresponding altitude.

Properties:

• Intersection Point: The altitudes intersect at a single point called the orthocenter, which can be located inside or outside the triangle.

• Each of the three altitudes in a scalene triangle has a different length due to the unequal sides and angles.

Altitude of an Isosceles Triangle

In an Isosceles Triangle, where 2 sides are of equal length, the altitude from the vertex angle (the angle between the two equal sides) to the base has special properties.

Properties:

• This altitude bisects the base into two equal segments.
• It also bisects the vertex angle, creating two congruent right triangles.
• In an isosceles triangle, this altitude is also a median and an angle bisector.

• Given an isosceles triangle with equal sides b and base a then altitude h is: $h=\sqrt{b^2-{\left(\frac{a}{2}\right)}^2}$

Altitude of an Equilateral Triangle

In an equilateral triangle, where all 3 sides and angles are equal, each altitude has unique properties due to the symmetry of the triangle.

Properties:

• It splits the opposite side into two equal segments.
• Each altitude also bisects the vertex angle into two 30-degree angles.
• Each altitude is also a median and an angle bisector.

• Given an equilateral triangle with side length a then altitude h is: $h=\frac{\sqrt{3}}{2}a$

Altitude of a Right Triangle

In a right triangle, one of the angles is 90 degrees. The altitudes from the two vertices forming the right angle are the legs of the triangle.

Properties:

• The legs are perpendicular to each other.
• The altitude extending from the right angle vertex to the hypotenuse is significant in various geometric calculations.
• The altitude extending from the right angle vertex to the hypotenuse divides the original triangle into two smaller right triangles, each similar to the original.

• In a right triangle with side lengths a and b for the legs, and hypotenuse c: $h=\frac{ab}{c}$​ where h is the altitude from the right-angle vertex to the hypotenuse.

Altitude of an Obtuse Triangle

In an obtuse triangle, one of the angles is greater than 90 degrees. The altitudes to the sides adjacent to the obtuse angle lie outside the triangle.

Properties:

• Each altitude is perpendicular to the opposite side.
• The orthocenter, the intersection point of the altitudes, lies outside the triangle.
• The altitudes from the vertices forming the obtuse angle are drawn to the extended opposite sides.

4.0Orthocenter of a Triangle

The orthocenter of a triangle is the point where all the three altitudes intersect. The orthocenter can lie inside, on, or outside the triangle, depending on the type of triangle.

Properties:

• The orthocenter is the common point where all three altitudes of a triangle meet.
• In an acute triangle, the orthocenter is located within the triangle.
• In a right triangle, the orthocenter is at the vertex of the right angle.
• In an obtuse triangle, the orthocenter is located outside the triangle.

5.0Difference between Median and Altitude of a Triangle

6.0Solved Example on Altitude of Triangle

Example 1: Find the altitude of a right triangle with legs of lengths a = 3 units and b = 4 units.

Solution:

In a right triangle, the altitude to the hypotenuse can be found using the formula:

$h=\frac{ab}{c}$, where c is the hypotenuse.

First, calculate the hypotenuse:

$c=\sqrt{a^2+b^2}=\sqrt{3^2+4^2}=\sqrt{9+16}=\sqrt{25}=5\text{ units }$

Now, calculate the altitude:

$h=\frac{3\times 4}{5}=\frac{12}{5}=2.4\text{ units }$

So, the altitude to the hypotenuse is 2.4 units.

Example 2: Find the altitude of an equilateral triangle with side length a = 6 units.

Solution:

In an equilateral triangle, the altitude can be found using the formula:

$h=\frac{\sqrt{3}}{2}a$

Substitute the side length:

$h=\frac{\sqrt{3}}{2}\times 6=3\sqrt{3}\approx 5.2\text{ units }$

So, the altitude of the equilateral triangle is approximately 5.2 units.

Example 3: Find the altitude of an isosceles triangle with equal sides b = 5 units and base a = 6 units.

Solution: In an isosceles triangle, the altitude to the base can be determined using the formula:

$h=\sqrt{b^2-{\left(\frac{a}{2}\right)}^2}$

Substitute the side lengths:

$h=\sqrt{5^2-{\left(\frac{6}{2}\right)}^2}=\sqrt{25-3^2}=\sqrt{25-9}=\sqrt{16}=4\text{ units }$

So, the altitude of the isosceles triangle is 4 units.

7.0Practice Question Based on Altitude of Triangle

1. Find the altitude to the hypotenuse of a right triangle with legs of lengths a = 5 units and b = 12 units.
2. Calculate the altitude of an equilateral triangle with side length a = 10 units.
3. Determine the altitude of an isosceles triangle with equal sides b = 10 units and base a = 12 units.