Lesson 36: Writing and Solving Proportions

New Concept

Proportions are equations stating that two ratios are equal, frequently used to solve mathematical problems involving similar figures and scale drawings.

What's next

Next, you'll explore worked examples on setting up and solving proportions with similar triangles. Soon, we'll tackle indirect measurement problems using shadows and scale drawings with blueprints and maps.

Property

If two geometric objects or figures are similar, they have the same shape but are not necessarily the same size. Corresponding angles are congruent ($\cong$), and the ratio of corresponding sides is equal. For $\triangle ABC \sim \triangle DEF$, $\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}$.

Examples

• Given $\triangle PQR \sim \triangle STU$, if $\angle P = 45^\circ$ and $\angle Q = 85^\circ$, then $\angle S = 45^\circ$ and $\angle T = 85^\circ$.
• If rectangle ABCD is similar to rectangle EFGH with $AB=4, BC=8$ and $EF=6, FG=12$, the side ratios are equal: $\frac{AB}{EF} = \frac{4}{6} = \frac{2}{3}$ and $\frac{BC}{FG} = \frac{8}{12} = \frac{2}{3}$.

Explanation

Imagine a photo and its enlargement—that's what similar figures are all about! They have the exact same shape but can be different sizes. This means all their corresponding angles are perfectly equal, and the ratio of their corresponding sides is always the same. It’s like making a perfect copy, just zoomed in or out.

Property

A scale factor is the ratio of a side length of a figure to the side length of a similar figure.

Examples

• If a small triangle has a side of length 6 and its similar, larger version has a corresponding side of length 18, the scale factor from small to large is $\frac{18}{6} = 3$.
• A model car is 5 inches long, while the real car is 15 feet (180 inches) long. The scale factor of the model to the real car is $\frac{5}{180} = \frac{1}{36}$.

Explanation

The scale factor is your secret resizing code! It’s the single number you multiply by to get from one similar figure to another. If you're enlarging a shape, the factor is greater than 1. If you're shrinking it, the factor is less than 1. It’s the ultimate copy-and-resize tool for perfect proportions!

Property

A scale drawing is a drawing that reduces or enlarges the dimensions of an object by a constant factor. The scale is a ratio showing the relationship between a scale drawing or model and the actual object.

Examples

• On a blueprint with a scale of 1 inch : 3 feet, a room that measures 5 inches wide is actually $5 \times 3 = 15$ feet wide.
• A world map uses a scale of 1:40,000,000. A 2 cm distance on the map represents an actual distance of $2 \times 40,000,000 = 80,000,000$ cm, which is 800 km.

Explanation

Ever used a map or seen building blueprints? That's a scale drawing in action! It's a miniature version of something huge, where every part is shrunk down perfectly in proportion. The scale, like '1 cm : 10 meters,' is the magic rule that lets you convert the drawing's measurements back to their real-world size.