High School Geometry Quiz: Applying Geometric Mean Concepts

📐 Quick Study Guide

• 📏 Geometric Mean Definition: The geometric mean of two positive numbers $a$ and $b$ is the positive number $x$ such that $\frac{a}{x} = \frac{x}{b}$. Therefore, $x = \sqrt{ab}$.
• ➗ Geometric Mean (Leg) Theorem: In a right triangle, the altitude to the hypotenuse divides the triangle into two smaller triangles that are similar to the original triangle and to each other. If the altitude is drawn to the hypotenuse, then each leg of the right triangle is the geometric mean between the hypotenuse and the segment of the hypotenuse that is adjacent to that leg.
• ✨ Geometric Mean (Altitude) Theorem: In a right triangle, the altitude to the hypotenuse divides the hypotenuse into two segments. The length of the altitude is the geometric mean between these two segments.
• 💡 How to find it: To find the geometric mean of a set of numbers, multiply them together and then take the $n$th root, where $n$ is the number of values in the set.

✍️ Practice Quiz

1. What is the geometric mean between 4 and 9?
   1. 2
   2. 6
   3. 6.5
   4. 36
2. In a right triangle, the altitude to the hypotenuse divides it into segments of length 3 and 12. What is the length of the altitude?
   1. 4
   2. 6
   3. 9
   4. 15
3. If the geometric mean between two numbers is 8, and one of the numbers is 2, what is the other number?
   1. 4
   2. 16
   3. 32
   4. 64
4. In right triangle $ABC$, with right angle at $C$, altitude $CD$ is drawn to hypotenuse $AB$. If $AD = 4$ and $AB = 16$, find $AC$.
   1. $4\sqrt{3}$
   2. $4\sqrt{5}$
   3. 8
   4. $8\sqrt{3}$
5. Find the geometric mean of the numbers 2, 4, and 8.
   1. 4
   2. $\sqrt[3]{64}$
   3. $\sqrt{32}$
   4. $\sqrt[3]{14}$
6. The sides of a rectangle are 5 and 20. What is the side length of a square with the same area as the rectangle?
   1. 10
   2. 12.5
   3. 25
   4. 100
7. In a right triangle, the length of one leg is 6 and the length of the hypotenuse is 10. What is the geometric mean between the hypotenuse and the segment of the hypotenuse adjacent to that leg if the altitude is drawn to the hypotenuse?
   1. 3.6
   2. 4.8
   3. 6
   4. 8