Math Problem Statement

These figures are similar. The area of one is given. Find the area of the other. The area of the larger figure is 196 in², and the side lengths are 15 inches and 21 inches for the smaller and larger figure, respectively.

Solution

The problem involves two similar figures, where the area of one figure is given, and the task is to find the area of the other figure. We can solve this using the concept of similarity in geometry.

Step-by-step solution:

  1. Recognize that the figures are similar: Since the figures are similar, the ratio of their areas is the square of the ratio of their corresponding side lengths.

  2. Find the ratio of the side lengths: The side lengths of the first figure (smaller one) is 15 inches, and the side length of the second figure (larger one) is 21 inches.

    Side length ratio=2115=75\text{Side length ratio} = \frac{21}{15} = \frac{7}{5}

  3. Square the ratio of the side lengths to get the ratio of the areas: (75)2=4925\left(\frac{7}{5}\right)^2 = \frac{49}{25}

  4. Set up a proportion using the areas: Let the area of the smaller figure be AA (unknown), and the area of the larger figure is given as 196 in².

    A196=2549\frac{A}{196} = \frac{25}{49}

  5. Solve for AA: Multiply both sides of the equation by 196: A=196×2549=196×5272=100in2A = 196 \times \frac{25}{49} = 196 \times \frac{5^2}{7^2} = 100 \, \text{in}^2

Thus, the area of the smaller figure is 100 square inches.


Would you like a more detailed breakdown of any step, or do you have any specific questions?

Here are 5 related questions to consider:

  1. What happens if the figures are not similar?
  2. How do we calculate the ratio of areas if only the perimeters are given?
  3. What other properties can be derived from similar figures?
  4. How can we use scale factors in three-dimensional geometry?
  5. How does the Pythagorean theorem apply to similar triangles?

Tip: When working with similar shapes, always remember that the ratio of areas is the square of the ratio of their corresponding sides.

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Math Problem Analysis

Mathematical Concepts

Geometry
Similar Figures
Ratio of Areas

Formulas

Side length ratio = larger side / smaller side
(Side length ratio)^2 = Ratio of areas
Area of smaller figure = (Area of larger figure) * (smaller side^2 / larger side^2)

Theorems

Similarity Theorem

Suitable Grade Level

Grades 8-10