Math Problem Statement

Enlarge the following geometric shape by a zoom factor of 2. What is the perimeter of the new figure? What is the area of the new figure?

Solution

To solve this problem, let's go step by step.

Step 1: Apply the Zoom Factor of 2

When a geometric shape is enlarged by a zoom factor of 2, each of its side lengths is multiplied by 2.

The original side lengths are:

  • 4, 3, 6, and 5.

After applying the zoom factor of 2:

  • 4×2=84 \times 2 = 8
  • 3×2=63 \times 2 = 6
  • 6×2=126 \times 2 = 12
  • 5×2=105 \times 2 = 10

Step 2: Calculate the Perimeter of the New Figure

The perimeter is the sum of all side lengths.

Perimeter=8+6+12+10=36\text{Perimeter} = 8 + 6 + 12 + 10 = 36

Thus, the perimeter of the new figure is 36 units.

Step 3: Calculate the Area of the New Figure

The shape resembles a trapezoid, with two parallel sides (3 and 6 originally) and height 4.

Original Area of the Trapezoid:

Using the formula for the area of a trapezoid:

Area=12×(b1+b2)×h\text{Area} = \frac{1}{2} \times (b_1 + b_2) \times h

Where:

  • b1=3b_1 = 3
  • b2=6b_2 = 6
  • h=4h = 4

Area=12×(3+6)×4=12×9×4=18square units\text{Area} = \frac{1}{2} \times (3 + 6) \times 4 = \frac{1}{2} \times 9 \times 4 = 18 \, \text{square units}

Area of the Enlarged Figure:

Since the enlargement factor is 2, the area will increase by 22=42^2 = 4 times.

New Area=18×4=72square units\text{New Area} = 18 \times 4 = 72 \, \text{square units}

Final Answers:

  • Perimeter of the new figure: 36 units.
  • Area of the new figure: 72 square units.

Do you need any further details or have any questions?
Here are some follow-up questions you might find helpful:

  1. How does the area scale with different zoom factors?
  2. How would the perimeter change with a zoom factor of 3 instead?
  3. What if the shape was rotated—would the perimeter or area change?
  4. Can you confirm if all shapes scale similarly with zoom factors?
  5. How would we calculate the side lengths if the original diagram had missing measurements?

Tip: Remember that area scales with the square of the zoom factor, while perimeter scales linearly!

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Math Problem Analysis

Mathematical Concepts

Geometry
Scaling
Perimeter
Area

Formulas

Perimeter = sum of all side lengths
Area of a trapezoid = (1/2) * (b1 + b2) * h

Theorems

Scaling Theorem for Perimeter and Area

Suitable Grade Level

Grades 6-8