Ratios & Cross-Multiplying
A ratio compares two quantities. A proportion says two ratios are equal. Cross-multiplying is the fundamental tool for solving proportions.
Introduction
Ratios pop up everywhere in geometry: scale factors, similar figures, trigonometric ratios, and more. Before we can work with similarity, we need to be fluent in setting up and solving proportions. The key technique is cross-multiplication.
Past Knowledge
Fractions & solving equations (Algebra). Proportional relationships (7th grade).
Today's Goal
Write ratios, set up proportions, and solve using cross-multiplication.
Future Success
Properties of proportions (7.1.2), dilations (7.1.3), similarity (7.2).
Key Concepts
Ratio
A ratio compares two quantities by division. The ratio of to can be written as:
Proportion & Cross-Multiplication
A proportion states that two ratios are equal. If , then:
This works because multiplying both sides by eliminates the denominators.
Extended Ratios
An extended ratio compares three or more quantities: . Use a single variable so that where is the ratio.
Worked Examples
Solving a Proportion
Solve:
Cross-multiply:
→
Variable on Both Sides
Solve:
Cross-multiply:
→ →
Extended Ratio — Triangle Angles
The angles of a triangle are in the ratio . Find each angle.
Let the angles be .
→ →
Angles:
— it's a right triangle!
Common Pitfalls
Ordering the Ratio Incorrectly
“The ratio of width to height is 3:5” means , NOT . The order matters!
Forgetting to Distribute
When cross-multiplying , the left side becomes , not .
Real-Life Applications
Recipe Scaling
If a recipe serves 4 and uses 3 cups of flour, how much flour for 10 people? Set up the proportion , cross-multiply, and solve.
Map Distances
A map scale of 1 inch : 50 miles lets you set up proportions to find real-world distances from map measurements.
Practice Quiz
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