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Golden Ratio

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Golden Ratio

Study Guide

Key Definition

The Golden Ratio is when a line segment is divided into two parts such that the ratio of the whole segment to the longer part is equal to the ratio of the longer part to the shorter part. Mathematically, $\frac{AB}{AC} = \frac{AC}{CB}$, where $\overline{AB}$, $\overline{AC}$, and $\overline{CB}$ are line segments.

Important Notes

The Golden Ratio is approximately $1.618033$.
It is denoted by the Greek letter $\phi$.
Found in human body proportions and nature.
Occurs in architecture and art.
Mathematically, it satisfies $x^2 = x + 1$.

Mathematical Notation

$\phi$ denotes the Golden Ratio

$\frac{AB}{AC} = \frac{AC}{CB}$ is the defining property

$x^2 = x + 1$ is the characteristic equation

Remember to use proper notation when solving problems

Why It Works

The Golden Ratio works because it creates aesthetically pleasing proportions. This ratio is found in the Fibonacci sequence and leads to naturally occurring patterns in plants and animals.

Remember

The Golden Ratio is often seen in nature and art. Its value is $\frac{1 + \sqrt{5}}{2}$.

Quick Reference

Value of the Golden Ratio:$\phi \approx 1.618033$

Understanding Golden Ratio

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Video explanation of this concept

Beginner

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Beginner Explanation

The Golden Ratio is a special number that appears when a line is divided so that the whole is to the longer part as the longer part is to the shorter part. Its value is approximately $1.618033$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

Given a line segment divided in the Golden Ratio, if the whole length $AB = 1$, what is the length of the longer part $AC$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

A teenager wants to design a poster with dimensions that follow the Golden Ratio. If the width is $24$ inches, what should the height be to follow the Golden Ratio?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

If a rectangle is divided into a square and a smaller rectangle, and the smaller rectangle is similar to the original, find the ratio of the sides of the rectangle.

Challenge Quiz

Single Choice Quiz

Advanced

A pentagon is inscribed in a circle. If the side of the pentagon is $s$, find the expression for the diagonal in terms of $s$ if it follows the Golden Ratio.

Recap

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