Math Problem Statement

Solve

Solution

To determine which triangles are similar by SSS (Side-Side-Side) similarity, we need to check if the corresponding sides of the triangles are proportional. That is, the ratios of the lengths of corresponding sides should be equal.

Let's analyze the triangles provided:

  1. Triangle ABC:

    • AB=4AB = 4
    • BC=6BC = 6
    • CA=8CA = 8
  2. Triangle DEF:

    • DE=6DE = 6
    • EF=9EF = 9
    • FD=12FD = 12
  3. Triangle GHI:

    • GH=6GH = 6
    • HI=6HI = 6
    • IG=10.5IG = 10.5

Comparing the Triangles:

  1. Comparing Triangle ABC and Triangle DEF:

    • ABDE=46=23\frac{AB}{DE} = \frac{4}{6} = \frac{2}{3}
    • BCEF=69=23\frac{BC}{EF} = \frac{6}{9} = \frac{2}{3}
    • CAFD=812=23\frac{CA}{FD} = \frac{8}{12} = \frac{2}{3}

    All three side ratios are equal, so Triangle ABC is similar to Triangle DEF by SSS similarity.

  2. Comparing Triangle ABC and Triangle GHI:

    • ABGH=46=23\frac{AB}{GH} = \frac{4}{6} = \frac{2}{3}
    • BCHI=66=1\frac{BC}{HI} = \frac{6}{6} = 1
    • CAIG=810.50.76\frac{CA}{IG} = \frac{8}{10.5} \approx 0.76

    The side ratios are not all equal, so Triangle ABC is not similar to Triangle GHI.

  3. Comparing Triangle DEF and Triangle GHI:

    • DEGH=66=1\frac{DE}{GH} = \frac{6}{6} = 1
    • EFHI=96=1.5\frac{EF}{HI} = \frac{9}{6} = 1.5
    • FDIG=1210.51.14\frac{FD}{IG} = \frac{12}{10.5} \approx 1.14

    The side ratios are not all equal, so Triangle DEF is not similar to Triangle GHI.

Answer:

The correct choice is ABC and DEF.

Do you want details or have any questions about the solution? Here are some related questions to further explore this topic:

  1. How can you determine the similarity of triangles using the SAS (Side-Angle-Side) criterion?
  2. If two triangles have one pair of proportional sides and one equal angle, can they be similar?
  3. What is the difference between similarity and congruence in triangles?
  4. How do changes in the proportions of side lengths affect the similarity of triangles?
  5. Can two triangles be similar if their corresponding side lengths are in different ratios?

Tip: When using SSS similarity, always ensure the ratios of corresponding sides are exactly equal; even slight differences indicate the triangles are not similar.

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Math Problem Analysis

Mathematical Concepts

Geometry
Triangle Similarity

Formulas

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Theorems

SSS Similarity Criterion

Suitable Grade Level

Grades 7-9