Mathematics – Heights and Distances
(Manipal MET – Science – Mathematics)
What you’ll learn
• How to find the height (altitude) of a triangle.
• How to compute various distances – between two points, a point and a line, and between two parallel lines.
• The key formulas that appear repeatedly in entrance‑exam questions.
1. Simple Explanation
| Concept | What it is | When it is used |
|---|---|---|
| Height (altitude) of a triangle | A segment drawn from a vertex perpendicular to the opposite side (or its extension). | To find the area of a triangle or to locate a point on the opposite side. |
| Distance between two points | The straight‑line length connecting them. | For coordinate‑geometry problems, or when you need the length of a side. |
| Distance from a point to a line | The perpendicular segment from the point to the nearest point on the line. | To check if a point lies on a line, or to solve optimisation problems. |
| Distance between two parallel lines | The length of the perpendicular segment that meets both lines. | To find separation between roads, building walls, etc. |
2. Key Points
2.1 Height of a Triangle
-
Using Area
[ \text{Area} = \tfrac12 \times \text{base} \times \text{height} \quad\Rightarrow\quad h = \frac{2,\text{Area}}{\text{base}} ] -
Right‑triangle Altitude (to the hypotenuse)
For a right triangle with legs (a, b) and hypotenuse (c):
[ h_{\text{alt}} = \frac{a,b}{c} ] -
By Similarity
In a right triangle, the altitude to the hypotenuse creates two smaller right triangles similar to the original.
[ \frac{h}{a} = \frac{b}{c} \quad\text{and}\quad \frac{h}{b} = \frac{a}{c} ] These give the same result as (2).
2.2 Distance Between Two Points (Cartesian Plane)
Let (P(x_1,y_1)) and (Q(x_2,y_2)).
[ d(P,Q) = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} ]
2.3 Distance from a Point to a Line
For line (Ax + By + C = 0) and point (P(x_0,y_0)):
[ d(P,\text{line}) = \frac{|A x_0 + B y_0 + C|}{\sqrt