## MATH100: Visualizing a Formula

Name:

Names of people you worked with:

This exercise is based on the one on page 34 of your book. In the
exercise, geometric areas are expressed using algebraic formulas and
are used to verify algebraic identities. This exercise will
familiarize you with the identities and also give you more insight
either into geometry or algebra.

The area of a rectangle with a length of a units and a height of b
units is (a b) square units.

The area of a rectangle with length a and height 2b is (2ab) square units -- the same as the area of two a by b rectangles combined.

- Use the space below to sketch a rectangle with length 2a and height 2b.

- What is the area of the rectangle you drew above?

- On page 34 of
*Precalculus: Mathematics for
Calculus* is a picture of a blue square (a special type of
rectangle) whose length is a+b units and whose height is a+b units.
What is the area of this square?

orange | purple |

pink | orange |

- To the right of the blue square is a subdivided square (sketched above). Find the
areas of:
- The orange rectangles:
- The purple square:
- The pink square:

- If you add the areas of the pieces in the right hand figure, you
should get the same value as the area of the left hand figure. Add
the areas you found in the previous question and compare the sum to
the area of the blue square. Are the two algebraic expressions
equivalent?

- What is the area of the large square on the left side of the
center of the page?

- What is the area of the small orange square on the left side of
the center of the page?

- Explain why the blue and pink regions on the left side of the center of the page combine to form an area of a² - b².

- The blue rectangle on the left side of the center of the page has
height (a - b) and width a. What are the dimensions of the pink
rectangle in this figure?

- The blue and pink rectangles on the right side of the center of the page have the same dimensions as those on the left of the page.
Use the formula for the area of a rectangle to find the area of the combined blue and pink rectangles on the right of the page.

- Explain how the left and right figures at the center of the page verify the formula a² - b² = (a + b)(a - b).

**Extra Credit:** Explain how the figure at the bottom of the page verifies the
formula (a + b)^{3} = a^{3} + 3a²b + 3ab² +
b^{3}.