Opposite
(KS3, Year 7)

The opposite is the shorter side opposite a given angle of a right triangle (also called a right-angled triangle). If we choose one of the angles of the right triangle that is not the right angle then the opposite is the shorter side opposite that angle. The image below shows what we mean by the angle (labelled θ) and the opposite:

The Opposite and Trigonometry

Trigonometry relates an angle within a right triangle to the lengths of its three sides (called the hypotenuse, adjacent and opposite). The image below shows what we mean by an angle (labelled θ) and the three sides (the hypotenuse, adjacent and opposite). right_triangle_sides_angle

The sine function relates the angle in a right triangle to the ratio of the length of the opposite side to the length of the hypotenuse:
The tangent function relates the angle in a right triangle to the ratio of the length of the opposite side to the length of the adjacent:

The Opposite and Pythagoras' Theorem

Pythagoras' theorem concerns the relationship between the lengths of the three sides of the right triangle. Pythagoras' theorem (or the Pythagorean theorem) states that:

The square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides.

It is easier to remember Pythagoras' theorem as a formula:

a² + b² = c²

This formula can be rearranged to show the length of the opposite in terms of the other two sides:

pythagoras formula opposite

In the formula, a is the length of the adjacent, b is the length of the opposite and c is the length of the hypotenuse. The image below shows what we mean: pythagoras_triangle_adjacent_opposite_hypotenuse

pythagoras_triangle_adjacent_opposite_hypotenuse

The Sides of a Right Triangle

The shorter two sides of a right triangle are called legs or catheti (singular cathetus). The longest side is called the hypotenuse.

The Adjacent and the Opposite

If we define one of the angles (not the right angle) within a right triangle, then we can give different names to the two shorter sides. If we choose the angle labelled θ, the side next to it is called the adjacent and the side opposite is called the opposite. right_triangle_adjacent_opposite_theta

However, the sides are only defined relative to that angle. If we choose the other angle (labelled Φ), then different sides are labelled adjacent and opposite. right_triangle_adjacent_opposite_phi