Right Triangle Theorems and Trigonometric Ratios
A right-angled triangle (or right triangle) is an elementary geometric polygon containing exactly one 90-degree right angle ($\gamma = 90^\circ$ or $\pi/2$ radians). The two sides forming the perpendicular right angle are the legs ($a$ and $b$), and the longest side opposite the right angle is the hypotenuse ($c$).
The Pythagorean Theorem
Attributed to ancient Greek mathematician Pythagoras, the relationship between the lengths of the sides states:
$$a^2 + b^2 = c^2 \implies c = \sqrt{a^2 + b^2}$$
Trigonometric Functions (SOH CAH TOA)
- $$\sin(\alpha) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{a}{c}$$
- $$\cos(\alpha) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{b}{c}$$
- $$\tan(\alpha) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{a}{b}$$
Inradius and Circumradius Formulas
- Circumcircle Radius ($R$): The circumscribed circle passes through all three vertices. Its diameter equals the hypotenuse: $$R = \frac{c}{2}$$
- Incircle Radius ($r$): The radius of the largest circle tangent to all three interior sides is given by: $$r = \frac{a + b - c}{2} = \frac{ab}{a + b + c}$$
- Altitude to Hypotenuse ($h$): The perpendicular distance from the right angle vertex to the hypotenuse: $$h = \frac{ab}{c}$$
Frequently Asked Questions
How many values are needed to solve a right triangle?
Since one angle is always fixed at 90 degrees, you only need 2 known values to solve the rest of the triangle, as long as at least one of those values is a side length (e.g., two sides, or one side and one acute angle).
What is the formula for the hypotenuse of a right triangle?
According to the Pythagorean theorem, the hypotenuse c is calculated as c = sqrt(a² + b²), where a and b are the perpendicular leg lengths.
What is the altitude to the hypotenuse?
The altitude (height) perpendicular to the hypotenuse is calculated as h = (a * b) / c. It divides the right triangle into two smaller similar right triangles.
Can a right triangle have two 90-degree angles?
No. In Euclidean planar geometry, the sum of all interior angles of any triangle is exactly 180 degrees ($\alpha + \beta + \gamma = 180^\circ$). If one angle is 90 degrees, the other two acute angles must sum to 90 degrees ($\alpha + \beta = 90^\circ$), meaning neither can be 90 degrees.
What is a Pythagorean triple?
A Pythagorean triple consists of three positive integers $(a, b, c)$ that satisfy $a^2 + b^2 = c^2$. Common examples include (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25).