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

Verified step by step guidance

1. Identify the pairs of corresponding sides in the similar triangles. Since the triangles are similar, their corresponding angles are equal, and their corresponding sides are proportional.
2. Set up a proportion between the lengths of corresponding sides. For example, if side \(a\) in the first triangle corresponds to side \(a'\) in the second triangle, and side \(b\) corresponds to side \(b'\), then the ratio \(\frac{a}{a'} = \frac{b}{b'}\) holds.
3. Use the known side lengths to write an equation involving the unknown side length. Substitute the known values into the proportion to create an equation.
4. Solve the equation for the unknown side length by cross-multiplying and isolating the variable.
5. Check your solution by verifying that the ratios of all pairs of corresponding sides are equal, confirming the triangles remain similar.

Key Concepts

Similarity of Triangles

Two triangles are similar if their corresponding angles are equal and their corresponding sides are in proportion. This means the shape is the same but the size may differ, allowing us to relate side lengths using scale factors.

Ratio of Corresponding Sides

In similar triangles, the lengths of corresponding sides are proportional. This ratio, or scale factor, can be used to find unknown side lengths by setting up and solving proportions between known and unknown sides.

Setting Up and Solving Proportions

To find unknown sides, write a proportion equating the ratio of known sides in one triangle to the corresponding sides in the other. Solving this proportion involves cross-multiplication and algebraic manipulation to isolate and calculate the unknown length.