Question

Critical Thinking The geometric mean between two positive numbers \(a\) and \(b\) is the positive number \(x\) in \(\frac{a}{x}=\frac{x}{b}\). a. Solve \(\frac{4}{x}=\frac{x}{9}\) to find the geometric mean between 4 and 9 . b. In any right triangle \(X Y Z\), if \(\overline{Y W}\) is an altitude to the hypotenuse, then \(X Y\) is the geometric mean between \(X Z\) and \(X W\), and \(Y Z\) is the geometric mean between \(X Z\) and \(W Z\). If \(X Z=16\) and \(X W=4\), find \(X Y\).

Short answer

a. The geometric mean is 6. b. The length \( X Y \) is 8.

Step-by-step solution

Set up the Equation for Part a

We need to find the geometric mean \( x \) between the numbers 4 and 9 using the equation \( \frac{4}{x} = \frac{x}{9} \).

Cross Multiply and Solve for x

Cross multiplying the equation \( \frac{4}{x} = \frac{x}{9} \) gives \( 4 \cdot 9 = x \cdot x \), or \( 36 = x^2 \). Solving for \( x \), we take the square root of both sides, yielding \( x = 6 \).

Solve for Part b using the Geometric Mean Theorem

Given that \( X Y \) is the geometric mean between \( X Z = 16 \) and \( X W = 4 \), set up the equation: \( \frac{X Z}{X Y} = \frac{X Y}{X W} \).

Set up the Equation Using Given Values

Substitute the known values into the equation: \( \frac{16}{X Y} = \frac{X Y}{4} \).

Cross Multiply and Solve for X Y

Cross multiply to get \( 16 \cdot 4 = X Y \cdot X Y \), i.e., \( 64 = X Y^2 \). Solving for \( X Y \), take the square root of both sides to obtain \( X Y = 8 \).

Key concepts

Right Triangle

A right triangle is a type of triangle that has one angle measuring exactly 90 degrees. This specific angle is called the right angle, and the sides forming this angle are known as the legs of the triangle. The side opposite the right angle is the hypotenuse, which is the longest side.
If you visualize a ladder resting against a wall, the wall and the ground form a right angle, and the ladder is the hypotenuse.
You might hear students referring to the Pythagorean Theorem when they talk about right triangles. This important theorem helps in determining the length of a side of a right triangle when the other two sides are known.

Geometric Mean Theorem

The Geometric Mean Theorem is particularly unique and applies to right triangles. It offers a fascinating way to relate different segments drawn within or around a right triangle. The theorem states that in a right triangle, an altitude drawn from the right angle to the hypotenuse divides the triangle into two smaller, similar right triangles. Each of these smaller triangles maintains similarity with the original triangle.
Here's how it relates to the geometric mean:

• The altitude of the hypotenuse is the geometric mean of the two segments of the hypotenuse.
• Each leg of the original right triangle can be viewed as the geometric mean between the hypotenuse and the segment directly under that leg formed by the altitude's intersection with the hypotenuse.

This theorem helps build a deeper understanding of the relationships between the parts of the right triangle, making calculations for unknown sides more systematic.

Cross Multiplication

Cross multiplication is a method used to solve equations that involve fractions, specifically proportions. To find a missing variable, you essentially multiply the outer terms of the proportion by the inner terms of the proportion. This is called cross multiplying.
Consider an example proportion: \( \frac{a}{x} = \frac{b}{y} \). Cross multiplication will give you an equation \( a \cdot y = b \cdot x \).
This technique is incredibly useful for solving geometric mean problems, where often an equation of proportions arises. Once the cross multiplication is completed, the result is typically a simple algebraic equation that can be solved for the unknown variable.

Square Root Calculation

Calculating a square root is often a necessary mathematical skill, especially when dealing with geometric means or areas. The square root of a number \( x \) is a value that, when multiplied by itself, gives \( x \).
Using our examples from the Geometric Mean Theorem, when you find an equation like \( x^2 = a \), you'll want to solve for \( x \). To do this, you take the square root of both sides. If \( a = 64 \), then the square root of 64 is 8, since \( 8 \times 8 = 64 \).
Remember, finding the square root of a number is the reverse operation of squaring a number. It's crucial in solving geometric mean problems as it gives the correct length of the geometric mean.