8.7 Solve Proportion and Similar Figure Applications

Proportions are powerful tools for solving real-world problems involving ratios and scaling. They allow us to compare quantities and find unknown values by setting up equations with equivalent fractions. This skill is essential for many practical applications.

From recipe adjustments to indirect measurements, proportions help us tackle a wide range of scenarios. By mastering the techniques of cross-multiplication and algebraic manipulation, we can confidently solve proportions and apply them to real-life situations.

Solving Proportions

Clearing fractions in proportions

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• Proportions equate two ratios expressed as fractions ($\frac{2}{3} = \frac{4}{6}$)
• Cross-multiply terms to clear fractions multiply numerator of first ratio by denominator of second ratio and vice versa
• Set resulting cross-multiplied terms equal to each other
• Isolate variable using algebra techniques combine like terms, divide both sides by variable's coefficient
• Solve $\frac{x}{4} = \frac{3}{2}$ for $x$ cross-multiply $x \cdot 2 = 4 \cdot 3$, simplify $2x = 12$, divide by 2 $x = 6$

Applying Proportions

Proportions for real-world ratios

• Proportions solve problems with equivalent ratios or scaling
• Identify known and unknown quantities (variable) in problem
• Set up proportion with ratios comparing same types of quantities
• Solve proportion for unknown variable using cross-multiplication and algebra
• Recipe 2 cups flour per 3 cups sugar, 9 cups sugar available, find flour needed let $x$ be flour, set up $\frac{2}{3} = \frac{x}{9}$, cross-multiply and solve $2 \cdot 9 = 3 \cdot x$, $18 = 3x$, $x = 6$, need 6 cups flour
• Use proportional reasoning to solve real-world problems involving ratios and rates

Similar triangles for indirect measurement

• Similar triangles have same shape, different sizes congruent corresponding angles, proportional corresponding sides
• Find unknown lengths identify similar triangles, set up proportion with corresponding side ratios, solve for unknown length
• 6-foot person casts 4-foot shadow, nearby tree casts 12-foot shadow, find tree height let $x$ be tree height, set up $\frac{6}{4} = \frac{x}{12}$, cross-multiply and solve $6 \cdot 12 = 4 \cdot x$, $72 = 4x$, $x = 18$, tree is 18 feet tall
• Geometric similarity extends to other shapes beyond triangles, maintaining proportional relationships between corresponding parts

Scale and Ratio Applications

• Scale is a ratio that relates measurements on a model or map to actual measurements
• Use ratios to convert between scaled representations and real-world dimensions
• Apply proportions to solve problems involving maps, blueprints, and models