Solve right triangles with hypotenuse, legs, area, perimeter, altitude, and acute angles using trigonometry and the Pythagorean theorem.
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A right-angled triangle is a triangle in which one interior angle is exactly 90 degrees (a right angle). The side opposite the right angle is called the hypotenuse, and the other two sides are called legs or catheti.
c = \sqrt{a^2 + b^2}, \quad A = \frac{1}{2}ab, \quad h_c = \frac{ab}{c}, \quad \alpha = \arctan\left(\frac{a}{b}\right)Hypotenuse c is found using Pythagorean theorem. Area is half base times height. Altitude to hypotenuse is the product of legs divided by hypotenuse.
Scenario: Find hypotenuse and area for legs a = 6 and b = 8
Hypotenuse: c = √(6² + 8²) = √(36 + 64) = √100 = 10
Area: A = ½ × 6 × 8 = 24
Perimeter: P = 6 + 8 + 10 = 24
Angle α = arctan(6/8) ≈ 36.87°, Angle β = 90° - 36.87° = 53.13°
The hypotenuse is the side opposite the 90-degree right angle and is always the longest side of a right-angled triangle.
Use inverse trigonometric functions: angle α = arctan(opposite / adjacent) or arcsin(opposite / hypotenuse).
It is the perpendicular distance from the 90° vertex to the hypotenuse: h = (a × b) / c.
Special right triangles include 45°-45°-90° (isosceles right triangle, ratio 1:1:√2) and 30°-60°-90° (ratio 1:√3:2).
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