Problem 27 - Prealgebra

Question

CHALLENGE Triangle \(A B C\) has side lengths of 3 inches, 5 inches, and 6 inches. Triangle \(D E F\) has side lengths 4 inches, 6 inches, and 8 inches. Determine whether \(\triangle A B C \sim \triangle D E F .\) Explain.

Step-by-Step Solution

Answer

Triangles \(ABC\) and \(DEF\) are not similar because their side length ratios are not all equal.

Step 1: Define Similarity Criterion

Two triangles are similar if their corresponding angles are equal or if the ratios of the lengths of their corresponding sides are equal. A common method is to check the Side-Side-Side (SSS) similarity criterion, which requires that the corresponding side lengths of the two triangles are proportional.

Step 2: Determine Corresponding Sides and Ratios

Identify the corresponding sides between triangles \(ABC\) and \(DEF\) and then calculate the ratios. For Triangle \( ABC \): Side lengths are 3, 5, and 6 inches. For Triangle \( DEF \): Side lengths are 4, 6, and 8 inches.

Step 3: Calculate Ratios of Corresponding Sides

Calculate the ratios of corresponding side lengths: \(\frac{3}{4}, \frac{5}{6}, \text{and} \frac{6}{8} \). Simplify these fractions to \(0.75, 0.833,\text{and} 0.75\) respectively.

Step 4: Check for Equal Ratios

Compare the calculated ratios for similarity. If all three ratios are equal, the triangles are similar according to Side-Side-Side criterion. In this case, the ratios \(0.75, 0.833,\text{and} 0.75\) are not equal.

Step 5: Conclude Similarity

Since not all the corresponding side ratios are equal, triangles \(ABC\) and \(DEF\) are not similar.

Key Concepts

Side-Side-Side criterion

The Side-Side-Side (SSS) criterion is a fundamental rule in geometry used to determine the similarity of two triangles. According to this criterion, if the corresponding side lengths of two triangles are proportional, then the triangles are considered similar. By comparing the three pairs of corresponding side lengths and ensuring they maintain a consistent ratio, we can conclude that the triangles have the same shape though they might differ in size. This consistency demonstrates that their angles are identical, confirming triangle similarity through SSS.

For example, in the triangles provided, for SSS to hold, the ratios of their corresponding sides must be equal. In this problem, the side ratios calculated were not equal, and therefore, the triangles were not similar by the SSS criterion.

ratio of side lengths

Understanding the ratio of side lengths is crucial when examining triangle similarity. Ratio in this context is a way to compare the sizes of the sides of triangles. When we compare triangles, knowing how the lengths of one triangle relate to the lengths of another helps determine similarity. This means that if each side of one triangle is, say, twice as long as the corresponding side of another triangle, they could be similar.

The formula to calculate the ratio of the sides of two triangles is done by dividing the side length of one triangle by the corresponding side length of the other. In our example, we looked at the sides 3 from triangle ABC and 4 from triangle DEF, finding a ratio of \(\frac{3}{4}\), and so on. This information is vital in ensuring we are correctly assessing potential triangle similarity.

triangle properties

Triangles have unique properties that define their structure. First, understanding what makes a triangle similar is important — similarity depends on both angles and side ratios. If angles are equal or if side ratios hold constant across corresponding sides, triangles may be similar.

Other essential properties include the sum of internal angles, which is always 180 degrees, and the possibility of being scalene, isosceles, or equilateral based on side length equality. Within this context, even if two triangles share a set of properties (like being scalene), they are not necessarily similar unless they meet the criteria like the SSS.

geometry concepts

Geometry often involves deduction through known concepts. Among these, preserving proportions is a common idea, particularly with congruence and similarity. While congruence requires exact matches in both shape and size, similarity allows for a difference in scale under identical shape — a primary consideration of geometry.

Knowing how criteria like Angle-Angle (AA), Side-Angle-Side (SAS), and Side-Side-Side (SSS) work is essential in this field. These criteria help break down complex problems into solvable parts through ratios and properties. Every geometric proof rests on logical reasoning, making this subject a robust approach to understand the world's physical attributes.