Angle-based special right triangles have three angles that can be represented as integers. [1] X Research source There are only two angle-based special right triangles: the 30-60-90 triangle, and the 45-45-90 triangle. [2] X Research source Side-based special right triangles have three sides that can be represented as integers. [3] X Research source Each group of three integers is called a Pythagorean triple. There are an infinite number of Pythagorean triples—which means there are also an infinite number of side-based special right triangles.
Example: If the shortest side is 10 cm long, we can say that x=10 cm. We can calculate the length of two remaining sides by multiplying their coefficients by 10 cm: 10cm√3 and 210cm (√3 = roughly 1. 732). This gives us a triangle with sides equal to 10 cm, 17. 32 cm, and 20 cm.
Example: If x=50 m, then we know that two side of the triangle are 50 m long. To calculate the length of the remaining side, we multiple the coefficient √2 by 50 m (√2 = roughly 1. 414). This gives us a triangle with sides equal to 50 m, 50 m, and 70. 7 m.
Common Pythagorean triples: 5-12-13, 7-24-25, 8-15-17, 20-21-29, and 28-45-53. There are infinitely many more.
The numbers 3-4-5 represent the ratio of the sides with respect to each other. These ratios are 3/4, 3/5, 4/3, 4/5, 5/3, and 5/4. If, for example, you double the sides to 6-8-10, the ratios remain the same: 6/8 = 3/4, 6/10 = 3/5, and so on. Therefore, a 6-8-10 triangle is equivalent to the 3-4-5 triangle.
Remember: the numbers 20-21-20 represent the ratios of the sides with respect to each other.
