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 a² + b² = c².

The Pythagorean Theorem applies only to right triangles.

Using the Pythagorean Theorem: c² = 6² + 8² = 36 + 64 = 100, so c = √100 = 10 units.

The formula for finding the hypotenuse is a² + b² = c².

In the Pythagorean Theorem, 'a' and 'b' represent the lengths of the two legs of the right triangle, while 'c' represents the length of the hypotenuse.

The Pythagorean Theorem can be used in various real-life applications such as construction, navigation, and determining distances between points.

Using the Pythagorean Theorem: c² = a² + b², so 13² = 5² + b². This gives 169 = 25 + b², thus b² = 144, and b = 12 units.

In a 30-60-90 triangle, the lengths of the sides are in the ratio 1:√3:2, where 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 √2 times the length of each leg.

To determine if a triangle is a right triangle, check if the squares of the lengths of the two shorter sides add up to the square of the length of the longest side.