1. What is a Right-Angled Triangle?
A right-angled triangle is a triangle in which one of the angles is exactly \(90^\circ\). This angle is called a right angle. Because of this special angle, the triangle has unique side names and properties.
A
/|
/ |
/ | (Right angle at B)
C---B
2. Definition of a Right-Angled Triangle
Definition: A triangle is called a right-angled triangle if one of its interior angles is exactly \(90^\circ\).
Symbolically, if the right angle is at vertex \( B \), then:
\( \angle B = 90^\circ \)
The side opposite the right angle has a special name: the hypotenuse.
3. Special Sides of a Right-Angled Triangle
In a right-angled triangle, each side has a specific role:
- Hypotenuse: the side opposite the right angle; it is the longest side.
- Adjacent side: one of the sides forming the right angle.
- Opposite side: the other side forming the right angle.
3.1. Hypotenuse
The hypotenuse is the most important side of a right-angled triangle. In \( \triangle ABC \), if \( \angle B = 90^\circ \):
Hypotenuse = side \( AC \)
It is the longest side of the triangle.
4. Properties of a Right-Angled Triangle
- One angle is exactly \(90^\circ\).
- The side opposite the right angle (hypotenuse) is the longest side.
- The remaining two angles are always acute (less than \(90^\circ\)).
- The triangle satisfies the Pythagoras Theorem.
- The orthocentre of a right-angled triangle is the vertex at which the right angle is formed.
4.1. Angle Distribution
If one angle is \(90^\circ\), the sum of the remaining two angles must be:
\( 180^\circ - 90^\circ = 90^\circ \)
Thus, the two non-right angles are always acute.
5. Using Pythagoras Theorem
In a right-angled triangle, the Pythagoras Theorem states:
\( \text{Hypotenuse}^2 = \text{(Opposite side)}^2 + \text{(Adjacent side)}^2 \)
If the sides are \( a \), \( b \), and \( c \) where \( c \) is the hypotenuse, then
\( c^2 = a^2 + b^2 \)
This theorem is a foundation of geometry and is used extensively in trigonometry and coordinate geometry.
6. Examples of Right-Angled Triangles
Right-angled triangles appear everywhere around us:
- The corner of a book or room forms a right angle.
- Footpaths or roads branching at 90°.
- A ladder leaning against a wall forms a right-angled triangle with the ground.
- Triangles used in construction tools like set squares.
- Many roof supports use right-angled triangular frames.
7. Importance of Right-Angled Triangles
Right-angled triangles play a crucial role in mathematics because they connect geometry, trigonometry and coordinate geometry. Many real-life structures like bridges and ramps are designed using right-angled triangle principles.
The concept of the hypotenuse also leads to special triangles like 30–60–90 and 45–45–90 triangles studied later.