8.7 Solve Proportion and Similar Figure Applications

Solving Proportions

A proportion is an equation that says two ratios (fractions) are equal, like $\frac{2}{3} = \frac{4}{6}$. Solving proportion problems means finding an unknown value that keeps two ratios equivalent. This comes up constantly in real-world situations: scaling recipes, reading maps, and measuring objects indirectly.

Clearing Fractions in Proportions

The main technique for solving proportions is cross-multiplication. When you have a proportion like $\frac{a}{b} = \frac{c}{d}$, you multiply diagonally and set the results equal: $a \cdot d = b \cdot c$.

Here's the process step by step:

1. Write the proportion with the unknown variable in one of the four positions
2. Cross-multiply: multiply the numerator of each fraction by the denominator of the other
3. Set the two products equal to each other
4. Solve for the variable using basic algebra (combine like terms, divide both sides by the coefficient)

Example: Solve $\frac{x}{4} = \frac{3}{2}$ for $x$.

1. Cross-multiply: $x \cdot 2 = 4 \cdot 3$
2. Simplify: $2x = 12$
3. Divide both sides by 2: $x = 6$

You can check your answer by plugging it back in: $\frac{6}{4} = \frac{3}{2}$. Simplify the left side and you get $\frac{3}{2} = \frac{3}{2}$. It checks out.

Applying Proportions

Proportions for Real-World Ratios

When a word problem involves two quantities that scale together at a constant rate, you can solve it with a proportion. The key is making sure you set up your two fractions so that the same types of quantities are in the same positions.

Steps for setting up a word problem:

1. Identify the two quantities being compared and figure out which value is unknown
2. Write one ratio using the known pair of values
3. Write a second ratio using the unknown variable and the other known value
4. Make sure both fractions compare the quantities in the same order (e.g., flour on top, sugar on bottom in both fractions)
5. Cross-multiply and solve

Example: A recipe calls for 2 cups of flour per 3 cups of sugar. You have 9 cups of sugar. How much flour do you need?

• Let $x$ = cups of flour needed
• Set up the proportion with flour over sugar in both fractions: $\frac{2}{3} = \frac{x}{9}$
• Cross-multiply: $2 \cdot 9 = 3 \cdot x$
• Simplify: $18 = 3x$
• Divide: $x = 6$

You need 6 cups of flour.

Similar Triangles for Indirect Measurement

Similar triangles have the same shape but different sizes. Their corresponding angles are equal, and their corresponding sides are proportional. This property lets you measure things you can't reach directly, like the height of a tree or a building.

Steps for solving similar triangle problems:

1. Identify the two similar triangles in the problem (often formed by an object and its shadow)
2. Match up the corresponding sides between the two triangles
3. Set up a proportion using two pairs of corresponding sides, with the unknown in one position
4. Cross-multiply and solve

Example: A 6-foot person casts a 4-foot shadow. A nearby tree casts a 12-foot shadow. How tall is the tree?

The person and their shadow form one triangle; the tree and its shadow form a similar triangle. The height corresponds to height, and shadow corresponds to shadow.

• Let $x$ = height of the tree
• Set up the proportion (height over shadow): $\frac{6}{4} = \frac{x}{12}$
• Cross-multiply: $6 \cdot 12 = 4 \cdot x$
• Simplify: $72 = 4x$
• Divide: $x = 18$

The tree is 18 feet tall. This technique works because the sun hits both objects at the same angle, creating similar triangles.

Geometric similarity isn't limited to triangles. Any two shapes with equal corresponding angles and proportional corresponding sides are similar, and you can use the same proportion method to find unknown lengths.

Scale and Ratio Applications

A scale is a specific ratio that relates a measurement on a model, map, or blueprint to the actual real-world measurement. For example, a map scale of $\frac{1}{50000}$ means 1 cm on the map equals 50,000 cm (or 500 m) in real life.

To solve scale problems, set up a proportion with the scale ratio on one side and the known/unknown measurements on the other, then cross-multiply and solve just like any other proportion.