WORLD OF SHAPES

21. RIGHT ANGLED TRIANGLE

Definition: A right-angled triangle has one angle measuring 90 degrees.
Pythagorean Theorem: In a right-angled triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides. Mathematically, this can be expressed as:

[

c^2 = a^2 + b^2

]

where (c) is the hypotenuse, and (a) and (b) are the other two sides.
Trigonometric Ratios: Right-angled triangles help define the basic trigonometric ratios:
- Sine ((\sin)): The ratio of the opposite side to the hypotenuse.
- Cosine ((\cos)): The ratio of the adjacent side to the hypotenuse.
- Tangent ((\tan)): The ratio of the opposite side to the adjacent side.
Special Types:
- Isosceles Right Triangle: Both legs are equal, and the angles are 45°, 45°, 90°.
- 30-60-90 Triangle: The lengths of the sides are in the ratio of (1:\sqrt{3}:2).
Area: The area of a right-angled triangle can be calculated using the formula:

[

\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}

]

Architecture and Construction: Right-angled triangles are essential in creating blueprints and ensuring structures are built at right angles.
Navigation: Used in determining distances and angles in navigation systems which rely on triangulation.
Physics: Many physical problems involve forces that can be broken down into components using right-angled triangles.
Computer Graphics: Triangles are used in rendering 3D models, and right-angled triangles simplify calculations for lighting and shading.
Surveying: Land surveys use right-angled triangles to calculate distances and angles on the terrain.

By understanding the properties and applications of right-angled triangles, we can see their relevance in numerous fields.