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.

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.

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.

• 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.

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.

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.

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.

Basic area formula:

For any triangle, area = \( \dfrac{1}{2} \times \text{base} \times \text{height} \).

In a right-angled triangle the two sides that form the right angle are perpendicular, so one can be taken as base and the other as height. If the legs are \( a \) and \( b \) then:

\( \text{Area} = \dfrac{1}{2}ab \)

Other forms:

• If \( c \) is the hypotenuse and \( h \) is the altitude from the right angle to the hypotenuse, then area = \( \dfrac{1}{2} c h \).
• Using trigonometry: if \( c \) is the hypotenuse and one acute angle is \( \theta \), then the legs are \( c\cos\theta \) and \( c\sin\theta \) so area = \( \dfrac{1}{2} c^2 \sin\theta\cos\theta = \dfrac{1}{4} c^2 \sin 2\theta \).

Student note: When asked to find the area, first identify which sides are perpendicular—those are your base and height. If you only know the hypotenuse and an angle, use trig to get the legs and then apply \(\tfrac12 ab\).

There are three common situations when finding a missing side in a right-angled triangle. Let the legs be \( a \), \( b \) and the hypotenuse be \( c \):

• Case 1: Find the hypotenuse when both legs are known
  Use the formula: \( c = \sqrt{a^2 + b^2} \).
• Case 2: Find a leg when the hypotenuse and one leg are known
  Use: \( a = \sqrt{c^2 - b^2} \) or \( b = \sqrt{c^2 - a^2} \).
• Case 3: Real-life problems
  These often involve ladders, heights & distances, diagonals of rectangles, ramps, etc. Draw the triangle first and identify the right angle before applying the theorem.

Student note: Always identify the hypotenuse first—it is the side opposite the right angle. Many mistakes happen because students accidentally treat a leg as the hypotenuse.

Pythagoras theorem is not just a formula—it is a tool used in many situations. Here are some common applications:

• Finding diagonal lengths: Diagonal of a rectangle of sides \( a \) and \( b \) is \( \sqrt{a^2 + b^2} \).
• Ladder problems: If a ladder of length \( c \) reaches a height \( h \) on a wall, the distance from the wall is \( \sqrt{c^2 - h^2} \).
• Coordinate Geometry: Distance between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is derived from Pythagoras: \( \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \).
• Construction & engineering: Ensuring structures are level by checking diagonals form a right-angled triangle.

Student note: In most exam problems, the right-angled triangle is hidden inside another shape. Draw it clearly before applying the theorem.

You can find unknown angles using trigonometry. In a right-angled triangle:

\( \sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} \)

\( \cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} \)

\( \tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} \)

To find \( \theta \), use inverse trigonometric functions:

\( \theta = \sin^{-1}(\frac{O}{H}) \)

\( \theta = \cos^{-1}(\frac{A}{H}) \)

\( \theta = \tan^{-1}(\frac{O}{A}) \)

Student note: Decide which ratio to use based on the sides you are given. If the hypotenuse is given, start with sine or cosine. If not, use tangent.