Solution of Triangles: Distance of orthocentre from the sides

Solution of Triangles: Distance of Orthocentre from the Sides

The orthocentre of a triangle is the point where the three altitudes of the triangle intersect. An altitude of a triangle is a perpendicular line segment from a vertex to the opposite side (or to the line that contains the opposite side). The distance of the orthocentre from the sides of a triangle is an important concept in the study of triangles, particularly in trigonometry and geometry.

Understanding the Orthocentre

Before we delve into the distances from the orthocentre to the sides of a triangle, let's understand some basic properties of the orthocentre:

The orthocentre lies inside the triangle for an acute-angled triangle.
For a right-angled triangle, the orthocentre is at the vertex where the right angle is.
For an obtuse-angled triangle, the orthocentre lies outside the triangle.

Distance of Orthocentre from the Sides

The distance of the orthocentre from the sides of a triangle can be calculated using the altitudes and the sides of the triangle. Let's denote the sides of the triangle as a, b, and c, and the corresponding altitudes as h_a, h_b, and h_c.

Formulas

The distances d_a, d_b, and d_c from the orthocentre H to the sides BC, AC, and AB respectively can be calculated using the following formulas:

d_a = \frac{2 \cdot Area \ of \ Triangle}{a}
d_b = \frac{2 \cdot Area \ of \ Triangle}{b}
d_c = \frac{2 \cdot Area \ of \ Triangle}{c}

The area of the triangle can be calculated using Heron's formula:

Area \ of \ Triangle = \sqrt{s(s-a)(s-b)(s-c)}

where s is the semi-perimeter of the triangle given by:

s = \frac{a + b + c}{2}

Table of Differences and Important Points

Property	Acute-Angled Triangle	Right-Angled Triangle	Obtuse-Angled Triangle
Position of Orthocentre	Inside the triangle	At the right angle	Outside the triangle
Distance to Side `BC`	`d_a`	`h_a` (altitude)	`d_a`
Distance to Side `AC`	`d_b`	`0` (orthocentre is on this side)	`d_b`
Distance to Side `AB`	`d_c`	`0` (orthocentre is on this side)	`d_c`

Examples

Let's consider an example to illustrate how to calculate the distance of the orthocentre from the sides of a triangle.

Example 1: Acute-Angled Triangle

Suppose we have an acute-angled triangle with sides a = 7, b = 8, and c = 9.

Calculate the semi-perimeter s:
```
s = \frac{7 + 8 + 9}{2} = 12
```

Calculate the area using Heron's formula:

Area = \sqrt{12(12-7)(12-8)(12-9)} = \sqrt{12 \cdot 5 \cdot 4 \cdot 3} = \sqrt{720} = 6\sqrt{20}

Calculate the distances d_a, d_b, and d_c:

d_a = \frac{2 \cdot 6\sqrt{20}}{7} \approx 3.46
d_b = \frac{2 \cdot 6\sqrt{20}}{8} \approx 3.02
d_c = \frac{2 \cdot 6\sqrt{20}}{9} \approx 2.68

Example 2: Right-Angled Triangle

For a right-angled triangle with sides a = 6 (base), b = 8 (perpendicular), and c = 10 (hypotenuse), the orthocentre is at the vertex of the right angle, so the distances to the sides AC and AB are 0, and the distance to side BC is the length of the altitude from the right angle, which is b = 8.

Example 3: Obtuse-Angled Triangle

Calculating the distances for an obtuse-angled triangle follows the same procedure as for an acute-angled triangle, but it is important to note that the orthocentre will be outside the triangle.

By understanding these concepts and formulas, students can effectively solve problems related to the distances of the orthocentre from the sides of a triangle in their exams.