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CAHSEE Math Exam: Tutoring Solution21 chapters | 211 lessons

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Lesson Transcript

Instructor:
*Joseph Vigil*

In this lesson, you'll review the properties of a rectangle. Then you'll learn how to find the perimeter of a rectangle. Afterward, you can test your knowledge with a brief quiz.

Many of us know what **rectangles** look like, but what exactly makes a rectangle a rectangle?

To start off, all rectangles have four sides. But that alone doesn't make a rectangle. If that were the case, this would be a rectangle:

That, however, is a trapezoid -- definitely not a rectangle because in a rectangle, the four angles are all right angles.

Also, a rectangle's opposite sides are congruent, and when we say **congruent**, we mean they're of equal size.

We show congruence by marking the sides of equal length with the same number of hash marks as each other.

So a square is a type of rectangle because it has four sides with opposite sides congruent, and all its angles are right angles.

A square, however, has all four sides the same length, or congruent with each other, so we mark them with the same number of hash marks.

The **perimeter** of a rectangle is equal to the sum of all the sides. However, since a rectangle's opposite sides are congruent, we only need to know the length and width.

We can write this in an equation this way:

*P* = *l* + *w* + *l* + *w*

where *P* is the perimeter, *l* is the length of the rectangle and *w* is its width. But instead of writing the *l* and *w* twice, we can simplify the equation like this:

*P* = 2*l* + 2*w*

What if we were given the following measurements?

This rectangle has a length of 6 inches and a width of 3 inches. We can still calculate the rectangle's perimeter because we know that the other two sides also measure three and six inches, respectively.

So, we plug in 6 for *l* and 3 for *w* in our equation, and we have

*P* = 2(6) + 2(3) = 18

This rectangle's perimeter is 18 inches.

What if we were given the measurements for these two sides?

The image tells us that this rectangle's length is 4 feet, but we know nothing about its width. Even though we're given two sides, they're not the length and the width, both of which we need to determine perimeter. So we can't calculate this rectangle's perimeter from the information given.

Sammy was late to football practice, so his coach is making him run around the entire field three times. The field, including the end zones and the practice area behind the end zones is 160 yards long and 53 yards wide. What's the total distance Sammy has to run?

Well, since Sammy is running along all sides of the rectangular field, we're dealing with perimeter here.

Let's make a diagram of the field:

This rectangle has a length of 160 yards and a width of 53 yards. We have the information we need to plug into our perimeter formula.

*P* = 2*l* + 2*w*

Plugging in 160 for *l* and 53 for *w,* we have:

*P* = 2(160) + 2(53)

Multiplying 2 times 160 gives us 320, and multiplying 2 times 53 gives us 106, so we now have:

*P* = 320 + 106

We add 320 plus 106 and get:

*P* = 426

The field's perimeter is 426 yards.

Since Sammy has to run around the field three times, we'll need to multiply the perimeter by 3:

426 x 3 = 1,278

Sammy has to run 1,278 yards! He'd better get started!

An ant entered a bedroom and wanted to explore, so he walked all around it. The room is 6 feet wide and 10 feet long. What's the distance the ant traveled?

Again, we have a rectangular space, and we're measuring the distance around it, so we need to find the room's perimeter.

If we made a diagram, it would look like this:

We have the rectangle's length and width, so we can use those values in our perimeter formula:

*P* = 2*l* + 2*w*

*P* = 2(10) + 2(6)

*P* = 20 + 12

*P* = 32

That little ant covered 32 feet while exploring the room!

Annie, who owns the house the ant is exploring, wants to add another bedroom to the house. She knows that, because of space limitations, the room can have a maximum perimeter of 44 feet, and she's limited to a width of 10 feet. How long can she make the room?

Well, our unknowns in the perimeter formula are perimeter, length and width. We have two out of those three values in this situation, perimeter and width, so let's plug them in and then solve for length:

*P* = 2*l* + 2*w*

44 = 2*l* + 2(10)

44 = 2*l* + 20

We now subtract 20 from both sides of the equation, so that it reads like this:

44 - 20 = 2*l* + 20 - 20

This leaves us with:

24 = 2*l*

We divide both sides by 2 to find the length:

24 / 2 = 2*l* / 2

12 = *l*

With a width of 10 feet, the room can be a maximum of 12 feet long.

As long as we have two of the three unknowns (perimeter, length and width), we can always solve for the third.

These are the **properties of a rectangle**:

- It has four sides.
- All four of its angles are right angles.
- Its opposite sides are congruent or equal.

The formula for a rectangle's perimeter is:

*P* = 2*l* + 2*w*

If we're going to find a rectangle's perimeter, we don't need the measurements of all four sides, but we do need the length and width. As long as we have two of the three unknowns (perimeter, length and width), we can always solve for the third.

- Rectangles must have four sides, four right angles, and opposite sides equal
- The perimeter of a rectangle = 2
*l*+ 2*w*

When you are done, you should be able to:

- Recall the properties of a rectangle
- State the equation for the perimeter of a rectangle
- Apply the formula for a rectangle's perimeter to solve problems

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10 in chapter 18 of the course:

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CAHSEE Math Exam: Tutoring Solution21 chapters | 211 lessons

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