AI Tutor
How did an ancient Greek mathematician measure the height of the Great Pyramid of Giza over 2,500 years ago without ever climbing it?
Around 600 BCE, Thales of Miletus realized that sunlight hits all objects nearby at the exact same angle, casting shadows that form miniature replica triangles.
Because these shapes have identical angles, they are similar triangles, meaning their matching side lengths grow and shrink by the exact same scale factor.
How do we turn matching side lengths into a simple algebraic equation that we can solve every time?

Setting Up Proportions

A proportion is an equation stating that two ratios (fractions) are equal, written in the form ab=cd\frac{a}{b} = \frac{c}{d}.
When we write the similarity statement β–³ABCβˆΌβ–³DEF\triangle ABC \sim \triangle DEF, the letter order tells you exactly which sides match: side ABAB corresponds to DEDE, and side BCBC corresponds to EFEF.
To solve for an unknown side in a proportion, we use cross-multiplication: multiplying both sides by the denominators clears the fractions, turning ab=cd\frac{a}{b} = \frac{c}{d} into aβ‹…d=bβ‹…ca \cdot d = b \cdot c.
What does this look like with actual numbers and an unknown variable?

Worked Example: Finding an Unknown Side

Suppose β–³ABCβˆΌβ–³DEF\triangle ABC \sim \triangle DEF, with side lengths AB=6AB = 6, BC=8BC = 8, and corresponding side DE=9DE = 9. We want to find the unknown length of side EFEF, labeled as xx.