Math Problem Statement

Select all triangles that are similar to the triangle shown in the first image. Note they may not be drawn to scale.

Solution

The task involves identifying which triangles are similar to the one given in the first image. Triangles are considered similar if their corresponding sides are proportional, meaning the ratios of corresponding side lengths are equal.

Let's compare the given triangle with the options:

The triangle in the first image has side lengths:

  • 19 (left side)
  • 20 (bottom side)
  • 27 (right side)

To determine similarity, we will compare the ratios of corresponding sides for each of the four triangles in the second image:

  1. For the top-left triangle (38, 10, 94.5):

    • The ratio of sides for comparison would be:
      • 38/19 = 2
      • 10/20 = 0.5
      • 94.5/27 = 3.5 Since these ratios are not equal, this triangle is not similar.
  2. For the top-right triangle (70, 40.5, 19):

    • The ratio of sides:
      • 70/19 ≈ 3.68
      • 40.5/20 = 2.025
      • 19/27 ≈ 0.70 These ratios are also not equal, so this triangle is not similar.
  3. For the bottom-left triangle (19, 10, 67.5):

    • The ratio of sides:
      • 19/19 = 1
      • 10/20 = 0.5
      • 67.5/27 = 2.5 The ratios are not equal, so this triangle is not similar.
  4. For the bottom-right triangle (10, 81, 66.5):

    • The ratio of sides:
      • 10/19 ≈ 0.53
      • 81/20 = 4.05
      • 66.5/27 ≈ 2.47 These ratios are also unequal, meaning this triangle is not similar.

Based on the comparisons, none of the triangles in the second image are similar to the one shown in the first image.

Would you like further clarification on any of the steps or the reasoning behind this?

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Math Problem Analysis

Mathematical Concepts

Triangle similarity
Proportionality
Ratio comparison

Formulas

Ratio = Corresponding side of triangle A / Corresponding side of triangle B

Theorems

Triangle similarity theorem

Suitable Grade Level

Grades 8-10