24-25 Geometric Mean Leg Theorem 1-4

Front

What is the Geometric Mean?

Back

The Geometric Mean of two numbers a and b is the square root of their product, calculated as √(a*b). It is used in various applications, including finding the average rate of growth.

Front

State the Geometric Mean Leg Theorem.

Back

In a right triangle, the length of the altitude to the hypotenuse is the geometric mean of the lengths of the two segments of the hypotenuse.

Front

If the segments of the hypotenuse are 6 and 8, what is the length of the altitude?

Back

The altitude is √(6*8) = √48 = 4√3, approximately 6.9.

Front

How do you find the length of a leg in a right triangle using the Geometric Mean Leg Theorem?

Back

If a leg of the triangle is x, and the segments of the hypotenuse are p and q, then x = √(p*q).

Front

Solve for x: If the segments of the hypotenuse are 4 and 9, what is x?

Back

x = √(4*9) = √36 = 6.

Front

What is the relationship between the altitude and the segments of the hypotenuse?

Back

The altitude is the geometric mean of the two segments of the hypotenuse.

Front

If the altitude is 5, what are the segments of the hypotenuse if they are in the ratio 2:3?

Back

Let the segments be 2x and 3x. Then, 5 = √(2x * 3x) = √(6x^2) => 25 = 6x^2 => x^2 = 25/6 => x = √(25/6).

Front

What is the formula to find the length of the hypotenuse in a right triangle?

Back

The length of the hypotenuse c can be found using the Pythagorean theorem: c = √(a^2 + b^2), where a and b are the lengths of the other two sides.

Front

If one leg of a right triangle is 3 and the other is 4, what is the hypotenuse?

Back

Using the Pythagorean theorem: c = √(3^2 + 4^2) = √(9 + 16) = √25 = 5.

Front

What is the significance of the Geometric Mean in real-world applications?

Back

The Geometric Mean is often used in finance to calculate average rates of return, in biology for growth rates, and in various fields to find central tendencies of multiplicative processes.