Math Problem Statement

Review what you have learned so far about similarity as you solve for x.

Review what you have learned so far about similarity as you solve for x.

Solution

In this image, two rectangles are shown, one smaller and one larger. The dimensions for both rectangles are given, with the smaller rectangle having a height of 3 and a width xx, while the larger rectangle has a width of 21 and a perimeter of 60.

To solve for xx, we will use the concept of similarity, as the rectangles have the same shape (i.e., their corresponding sides are proportional).

Step 1: Use the perimeter to find the height of the larger rectangle.

The perimeter PP of a rectangle is given by the formula: P=2×(length+width)P = 2 \times (\text{length} + \text{width}) For the larger rectangle, the perimeter is 60 and the width is 21. Let the height of the larger rectangle be hh.

60=2×(h+21)60 = 2 \times (h + 21)

Solving for hh: 60=2h+4260 = 2h + 42 6042=2h60 - 42 = 2h 18=2h18 = 2h h=9h = 9

So, the height of the larger rectangle is 9.

Step 2: Set up a proportion using the similarity of the rectangles.

Since the rectangles are similar, the ratio of their corresponding sides must be equal. The corresponding sides are the heights and widths of the rectangles.

The ratio of the heights is: 39=13\frac{3}{9} = \frac{1}{3}

Therefore, the ratio of the widths must also be: x21=13\frac{x}{21} = \frac{1}{3}

Step 3: Solve for xx.

Now, solve the proportion: x21=13\frac{x}{21} = \frac{1}{3}

Cross-multiply to find xx: 3x=213x = 21 x=213=7x = \frac{21}{3} = 7

So, x=7x = 7.

Final Answer:

The value of xx is 7.

Would you like more details or have any other questions?

Here are some related questions to deepen your understanding:

  1. How do we know the rectangles in this problem are similar?
  2. How can you use perimeter to find the dimensions of a rectangle?
  3. What other properties of similar shapes could you apply to solve for unknown values?
  4. How would the solution change if the perimeter of the larger rectangle were different?
  5. Can this method be used to solve problems involving triangles or other polygons?

Tip: When dealing with similar shapes, always check that the ratios of corresponding sides are consistent across the entire shape.

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Math Problem Analysis

Mathematical Concepts

Geometry
Similarity of Shapes
Proportions

Formulas

Perimeter of a rectangle: P = 2(length + width)
Proportion of similar figures: corresponding sides are proportional

Theorems

Similarity Theorem
Proportionality Theorem

Suitable Grade Level

Grades 7-9