Homeclass 10MathsIntroduction to TrigonometryHypotenuse

Hypotenuse

The hypotenuse is the longest side of a right-angled triangle, always positioned directly opposite the 90-degree angle. In trigonometry, it serves as a constant reference side used to determine primary trigonometric ratios like sine and cosine.

Key Concepts

It is the only side of a right triangle that does not change its identity when the angle of reference changes.

According to the Pythagoras Theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

In trigonometric ratios, the hypotenuse acts as the denominator for both sine (opposite divided by hypotenuse) and cosine (adjacent divided by hypotenuse).

Formula & Equation

Hypotenuse^2 = Base^2 + Perpendicular^2

Hypotenuse is the longest side opposite the right angle, Base is the adjacent side, and Perpendicular is the opposite side of the right triangle.

Common Misconceptions

Myth: The hypotenuse changes depending on which acute angle you are looking at in trigonometry.

Fact: Unlike the opposite and adjacent sides, which swap depending on the reference angle, the hypotenuse is always fixed as the side opposite the 90-degree angle.

Real World Applications

A ladder leaning against a vertical wall forms the hypotenuse of a right triangle with the ground and the wall.

A wheelchair ramp connecting the ground to a raised platform acts as the hypotenuse.

Frequently Asked Questions

How do you find the length of the hypotenuse?

You can find it using the Pythagoras Theorem by taking the square root of the sum of the squares of the other two sides, or by using trigonometric ratios if an angle and one side are known.

Is the hypotenuse always the longest side of a triangle?

Yes, in a right-angled triangle, the hypotenuse is always the longest side because it is opposite the largest angle, which is 90 degrees.

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