Pythagorean Theorem
Understand the theorem, special right triangles, and the converse theorem
Overview
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs.
Therefore, the hypotenuse can be calculated as:
Here, \(a\) and \(b\) are the legs adjacent to the right angle and \(c\) is the hypotenuse.
Rearranging the equation also allows calculating a leg if the other leg and the hypotenuse are known.
Angle bisector
Median
Altitude and the Orthocenter
Triangle bisector
Pythagorean theorem
Pythagorean triples
Circle
Square
Rectangle
Rhombus
Parallelogram
Trapezoid
Example
Compute the hypotenuse
A triangle with \(a = 3\) and \(b = 4\):
Integer triples \((a,b,c)\) that satisfy the theorem are called Pythagorean triples.
Special Right Triangles
Two right-triangle types are especially important due to fixed side ratios.
30°, 60° and 90° Triangle
The side lengths follow the ratio \(b=a\cdot\sqrt{3}\), and the hypotenuse is twice the side opposite 30°.
Formulas
45°, 45° and 90° Triangle
The two legs are equal in length.
- The ratio of each leg to the hypotenuse is \(\displaystyle \frac{1}{\sqrt{2}}\)
- So \(\displaystyle a=\frac{c}{\sqrt{2}}\)
Converse of the Theorem
The converse helps classify triangles as acute, right, or obtuse.
Acute
If \(c^2 < a^2 + b^2\), then the triangle is acute.
Right
If \(c^2 = a^2 + b^2\), then the triangle is right-angled.
Obtuse
If \(c^2 > a^2 + b^2\), then the triangle is obtuse.
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