What are properties of right traingles

Answer 1

Right triangles, also known as right-angled triangles, possess several key properties:

1. Right Angle: One of the angles in a right triangle measures exactly 90 degrees. This is the defining characteristic of a right triangle.

2. Pythagorean Theorem: The sides of a right triangle are related by the Pythagorean theorem: \( a^2 + b^2 = c^2 \), where \( c \) is the length of the hypotenuse (the side opposite the right angle) and \( a \) and \( b \) are the lengths of the other two sides.

3. Trigonometric Ratios: The ratios of the lengths of the sides of a right triangle are used to define the primary trigonometric functions:

   • Sine: \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \)
   • Cosine: \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \)
   • Tangent: \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \)

4. Special Right Triangles: There are specific types of right triangles that have unique properties:

   • 45-45-90 Triangle: In this triangle, the two legs are equal, and the hypotenuse is \( \sqrt{2} \) times the length of a leg.
   • 30-60-90 Triangle: In this triangle, the sides are in the ratio \( 1:\sqrt{3}:2 \), with the side opposite the \( 30^\circ \) angle being the shortest.

5. Area: The area \( A \) of a right triangle can be calculated using the formula: \[ A = \frac{1}{2} \times \text{base} \times \text{height} \] where the base and height are the two legs of the triangle.

6. Circumscribed and Inscribed Circles: A right triangle can be inscribed in a circle (circumcircle) where the hypotenuse acts as the diameter. The radius of the circumcircle is half the length of the hypotenuse. Additionally, the inscribed circle (incircle) touches each side at exactly one point and can be calculated using the triangle's area and semi-perimeter.

7. Angle Properties: The sum of the angles in any triangle is \( 180^\circ \). Therefore, in a right triangle, the sum of the two acute angles is \( 90^\circ \).

These properties make right triangles an important subject in geometry and trigonometry, with numerous applications in various fields such as physics, engineering, and architecture.