Find the unknown side lengths in each pair of similar triangles. See Example 4.

Verified step by step guidance

1. Identify the corresponding sides of the similar triangles. Similar triangles have proportional sides, meaning the ratio of the lengths of corresponding sides is the same.
2. Set up a proportion using the known side lengths and the unknown side lengths. For example, if triangle A has sides a, b, c and triangle B has sides x, y, z, and you know a and x are corresponding sides, set up the proportion \( \frac{a}{x} = \frac{b}{y} = \frac{c}{z} \).
3. Choose one of the proportions to solve for the unknown side. For instance, if you need to find y, use \( \frac{b}{y} = \frac{a}{x} \).
4. Cross-multiply to solve for the unknown side. Using the example \( \frac{b}{y} = \frac{a}{x} \), cross-multiply to get \( b \cdot x = a \cdot y \).
5. Rearrange the equation to solve for the unknown side. Continuing the example, solve for y by dividing both sides by a: \( y = \frac{b \cdot x}{a} \).

Key Concepts

Similar Triangles

Similar triangles are triangles that have the same shape but may differ in size. This means that their corresponding angles are equal, and the lengths of their corresponding sides are proportional. Understanding the properties of similar triangles is essential for solving problems involving unknown side lengths, as it allows for the application of ratios to find missing measurements.

Proportionality

Proportionality in the context of similar triangles refers to the relationship between the lengths of corresponding sides. If two triangles are similar, the ratio of the lengths of any two corresponding sides is constant. This concept is crucial for setting up equations to solve for unknown side lengths, as it enables the use of cross-multiplication to find missing values.

Scale Factor

The scale factor is the ratio of the lengths of corresponding sides of two similar triangles. It indicates how much larger or smaller one triangle is compared to the other. Knowing the scale factor allows for the calculation of unknown side lengths by multiplying or dividing the known lengths by this factor, facilitating the solution of problems involving similar triangles.