Math Problem Statement

are these triangles similar?

Are these triangles similar?

Solution

To determine whether these triangles are similar, we can compare their corresponding sides and angles.

Key Similarity Criteria:

  1. AA (Angle-Angle): If two corresponding angles are equal, the triangles are similar.
  2. SSS (Side-Side-Side): If the ratios of all three corresponding sides are proportional, the triangles are similar.
  3. SAS (Side-Angle-Side): If two corresponding sides are proportional and the included angles are equal, the triangles are similar.

Step 1: Compare Corresponding Sides

The triangles GHF\triangle GHF and WUV\triangle WUV have these sides:

  • GF=48GF = 48, GH=42GH = 42, HF=36HF = 36
  • WV=48WV = 48, WU=50WU = 50, UV=39UV = 39

Ratios of Corresponding Sides:

  1. GFWV=4848=1\frac{GF}{WV} = \frac{48}{48} = 1
  2. GHWU=4250=0.84\frac{GH}{WU} = \frac{42}{50} = 0.84
  3. HFUV=36390.923\frac{HF}{UV} = \frac{36}{39} \approx 0.923

Since these ratios are not equal, the triangles do not have proportional sides.

Step 2: Check Angles

From the image, no information about the angles is given explicitly. Without knowing at least one pair of corresponding angles, we cannot check the AA or SAS similarity criteria.

Conclusion:

The triangles are not similar because their corresponding sides are not proportional, and no angle information supports similarity.

Would you like further explanation or a step-by-step breakdown of the similarity criteria?


Related Questions:

  1. What does it mean for triangles to be similar, and why is proportionality important?
  2. How do you calculate the ratios of corresponding sides in a triangle?
  3. Can triangles be similar if only one angle is equal between them?
  4. What is the significance of the SSS similarity criterion in geometry?
  5. How do real-world problems use the concept of similar triangles?

Tip:

When working with triangle similarity, always check for proportionality or equal angles first to determine the relationship between the triangles.

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Math Problem Analysis

Mathematical Concepts

Geometry
Triangle Similarity
Proportions

Formulas

Ratio of corresponding sides
Similarity criteria: AA, SSS, SAS

Theorems

Triangle Similarity Theorem

Suitable Grade Level

Grades 8-10