Pythagorean Theorem & Geometric Mean

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 can be expressed as: $$c^2 = a^2 + b^2$$ .

The geometric mean of two numbers a and b is given by the formula: $$\text{Geometric Mean} = \frac{\text{a} \times \text{b}}{\text{a} + \text{b}}$$ .

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

In a right triangle, the lengths of the sides must satisfy the Pythagorean Theorem: $$c^2 = a^2 + b^2$$ .

To determine if three lengths can form a right triangle, check if the square of the longest length equals the sum of the squares of the other two lengths.

Cross-multiplying gives: $$3 \times 4 = x^2$$ , thus $$x^2 = 12$$ , so $$x = 2\text{√}3$$ .

The area of a triangle is given by: $$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$$ . Thus, $$24 = \frac{1}{2} \times 8 \times h$$ , solving gives $$h = 6$$ .

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

Yes, because $$5^2 + 12^2 = 25 + 144 = 169 = 13^2$$ .

The geometric mean is calculated as: $$\text{Geometric Mean} = \text{√}(4 \times 16) = \text{√}(64) = 8$$ .