Pythagorean Theorem

The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). It is expressed as: $$a^2 + b^2 = c^2$$ .

Using the Pythagorean Theorem: $$3^2 + 4^2 = c^2$$ , which simplifies to $$9 + 16 = c^2$$ , so $$c^2 = 25$$ , thus $$c = 5$$ .

The distance between two points (x1, y1) and (x2, y2) in a coordinate plane is given by: $$d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2}$$ .

Using the Pythagorean Theorem: $$6^2 + b^2 = 10^2$$ , which simplifies to $$36 + b^2 = 100$$ , so $$b^2 = 64$$ , thus $$b = 8$$ .

In a 30-60-90 triangle, the lengths of the sides are in the ratio 1 : $$\sqrt{3}$$ : 2. The shortest side is opposite the 30-degree angle.

In a 45-45-90 triangle, the lengths of the legs are equal, and the hypotenuse is $$\sqrt{2}$$ times the length of each leg.

Yes, it is a right triangle because $$5^2 + 12^2 = 13^2$$ (25 + 144 = 169).

The converse states that if in a triangle, the square of the length of one side is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle.

The area of a right triangle can be found using the formula: $$Area = \frac{1}{2} \times base \times height$$ .

The perimeter of a right triangle is the sum of the lengths of all three sides: $$Perimeter = a + b + c$$ .