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CAHSEE Math Exam: Help and Review21 chapters | 239 lessons

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Lesson Transcript

Instructor:
*Yuanxin (Amy) Yang Alcocer*

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Did you know that a special type of rectangle can be found in the Mona Lisa painting? Learn what this type is. Also learn what makes a rectangle a rectangle, and how to make calculations with them.

Simply put, a **rectangle** is any four-sided figure that also has four right angles (90 degree angles). Look around you and you can see them all around you in the world. Most likely, the room you're in is some form of a rectangle or a combination of rectangles. It's a simple shape that is easy to work with.

You probably also drew a rectangle when you first drew a house as a kid. What shapes are most doors that you see? Aren't they rectangles? What about windows? Don't they all have four sides and four right angles?

In addition to having four sides and four 90-degree angles, there are some other properties that all rectangles must possess.

There are several things that make rectangles special and set them apart from other shapes. The first two have already been mentioned, but here they are again along with some more:

- It must have four sides.
- All four angles must be 90 degree right angles.
- A rectangle is a special case of a parallelogram; its opposite sides are parallel. Like a parallelogram, the opposite sides are equal in length to each other. There are two pairs of opposite sides, and each pair could have a different length, but each pair's sides will be equal to each other.
- The diagonals of a rectangle are equal in length to each other and they bisect each other at their point of intersection. When you draw a line that cuts the rectangle into two triangles and then do it again to the other two corners, these two lines will be equal in length to each other. These two lines also cross at exactly the midpoints of each. So, each diagonal cuts the other in half.

All of the above must be met for a shape to be considered a rectangle. Even with these requirements, there are rectangles that fall under their own very special types.

There are two special types of rectangles that have even stricter requirements than just rectangles.

- The first is a
**square**. A square is a rectangle with the added requirement that all sides are equal in length. You can fit a square in a rectangle whose width is the same as the width of the square, given that the length of the rectangle is longer than the width. - The second is the
**Fibonacci rectangle**. This special rectangle adds the requirement that the length to width ratio be 1.618. In other words, the length is 1.618 times longer than the width. So, if the width is 2, then the length is 2 times 1.618, or 3.236.

This special type of rectangle is also called the golden rectangle because its ratio is the golden ratio of 1.618. Looking at the Mona Lisa painting, mathematicians have noticed that the rectangle that goes from her head to her right hand and left elbow has the proportions of the golden rectangle.

When using rectangles in the real world to solve problems, there are only a few formulas to keep in mind. They are for the area, perimeter, and diagonals of the rectangle.

Just like other formulas, these are straightforward to use and require the plugging in of values to their appropriate places.

Let's say your business partner recently bought an office building and he wants to renovate it by putting in new flooring for the main meeting room. To figure out how much square footage of flooring he needs, he can use the formula for **area of a rectangle**.

If the room's dimensions are 20 feet by 30 feet, then he would require 20 * 30 = 600 square feet of flooring to perform the renovation.

If he wanted to add a baseboard all around the room, too, he can use the **perimeter formula**. This would give him 2(20) + 2(30) = 100 feet of baseboard required.

If he wanted to add a divider to the room so that it can be split into two triangular rooms, he can use the **diagonal formula** to figure out how long of a divider he needs.

Following the formula, he would first find 20^2 + 30^2 = 400 + 900 = 1,300 and then take the square root of it to get 36.05551275463989. Rounding that up, he would need approximately 36.06 feet for the divider.

To qualify as a **rectangle**, the shape must have four sides with four right angles whose opposite sides are parallel and equal in length to each other with diagonals of equal length and intersecting at their midpoints.

The three formulas that are needed for rectangles are the ones for **area**, **perimeter**, and **diagonal**.

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CAHSEE Math Exam: Help and Review21 chapters | 239 lessons

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