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Look at Rectangle ABCD below. Notice that it has been divided into 4 smaller rectangles. Given the area of three of the smaller rectangles, can you figure out the area of the fourth? In this concept, we will practice working with the dimensions of rectangles and area of rectangles.

Guidance

In order to solve the problem about the rectangle above, use the problem solving steps.

  • Start by describing what information is given.
  • Then, identify what your job is. In these problems, your job will be to figure out the area of the fourth rectangle.
  • Next, make a plan for how you will solve. In these problems, figure out the dimensions of the square first. Then figure out the dimensions of the other rectangles. Finally, find the area of the missing rectangle.
  • Then, solve the problem. Implement your plan.
  • Finally, check your solution. Verify the dimensions and area of each rectangle.

Example A

Rectangle EFGH is separated into 4 smaller rectangles.

Solution:

We can use problem solving steps to help.

& \mathbf{Describe:} && \text{The large rectangle contains} \ 4 \ \text{smaller rectangles. The areas of three of the}\\&&& \text{rectangles are given.} \ G \ \text{is a square. All dimensions are whole numbers.}\\\\& \mathbf{My \ Job:} && \text{Use the given areas. Figure out the area of Rectangle} \ E.\\\\& \mathbf{Plan:} && \text{Find common factors of the areas.} \ G \ \text{is a square, so the dimensions can be easily}\\&&& \text{determined. Figure out the dimensions of Rectangles} \ F \ \text{and} \ H. \ \text{This will give the}\\&&& \text{dimensions for Rectangle} \ E. \ \text{Use the area formula to figure out the area of}\\&&& \text{Rectangle} \ E.\\\\& \mathbf{Solve:} && G \ \text{is a square, so the dimensions are} \ 3 \ \text{in. by} \ 3 \ \text{in. That means that one of the}\\&&& \text{dimensions of Rectangle} \ F \ \text{is} \ 3 \ \text{inches. The area is} \ 12 \ \text{square inches, so the other}\\&&& \text{dimension must be} \ 12 \div 3, \ \text{or} \ 4 \ \text{inches. Likewise, one of the dimensions of}\\&&& \text{Rectangle} \ H \ \text{is} \ 3, \ \text{so the other is} \ 15 \div 3, \ \text{or} \ 5 \ \text{inches. The dimensions of Rectangle}\\&&& E \ \text{are} \ 4 \ \text{inches from the shared side with Rectangle} \ F \ \text{and} \ 5 \ \text{inches form the}\\&&& \text{shared side with Rectangle} \ H. \ \text{The area of} \ E \ \text{is} \ 4 \times 5, \ \text{or} \ 20 \ \text{square inches.}\\\\& \mathbf{Check:} && G: 3 \ \text{by} \ 3 \ \text{inches with an area of} \ 9 \ \text{sq in.}\\&&& F: 3 \ \text{by} \ 4 \ \text{inches with an area of} \ 12 \ \text{sq in.}\\&&& H: 5 \ \text{by} \ 3 \ \text{inches with an area of} \ 15 \ \text{sq in.}\\&&& E: 5 \ \text{by} \ 4 \ \text{inches with an area of} \ 20 \ \text{sq in.}

Example B

Rectangle JKLM is separated into 4 smaller rectangles.

Solution:

We can use problem solving steps to help.

& \mathbf{Describe:} && \text{The large rectangle contains} \ 4 \ \text{smaller rectangles. The areas of three of the}\\&&& \text{rectangles are given.} \ M \ \text{is a square. All dimensions are whole numbers.}\\\\& \mathbf{My \ Job:} && \text{Use the given areas. Figure out the area of Rectangle} \ K.\\\\& \mathbf{Plan:} && \text{Find common factors of the areas.} \ M \ \text{is a square, so the dimensions can be easily}\\&&& \text{determined. Figure out the dimensions of Rectangles} \ L \ \text{and} \ J. \ \text{This will give the}\\&&& \text{dimensions for Rectangle} \ K. \ \text{Use the area formula to figure out the area of}\\&&& \text{Rectangle} \ K.\\\\& \mathbf{Solve:} && M \ \text{is a square, so the dimensions are} \ 5 \ \text{in. by} \ 5 \ \text{in. That means that one of the}\\&&& \text{dimensions of Rectangle} \ L \ \text{is} \ 5 \ \text{inches. The area is} \ 40 \ \text{square inches, so the other}\\&&& \text{dimension must be} \ 40 \div 5, \ \text{or} \ 8 \ \text{inches. Likewise, one of the dimensions of}\\&&& \text{Rectangle} \ J \ \text{is} \ 5, \ \text{so the other is} \ 20 \div 5, \ \text{or} \ 4 \ \text{inches. The dimensions of Rectangle}\\&&& K \ \text{are} \ 4 \ \text{inches from the shared side with Rectangle} \ J \ \text{and} \ 8 \ \text{inches form the}\\&&& \text{shared side with Rectangle} \ L. \ \text{The area of} \ K \ \text{is} \ 4 \times 8, \ \text{or} \ 32 \ \text{square inches.}\\\\& \mathbf{Check:} && J: 4 \ \text{by} \ 5 \ \text{inches with an area of} \ 20 \ \text{sq in.}\\&&& K: 4 \ \text{by} \ 8 \ \text{inches with an area of} \ 32 \ \text{sq in.}\\&&& L: 5 \ \text{by} \ 8 \ \text{inches with an area of} \ 40 \ \text{sq in.}\\&&& M: 5 \ \text{by} \ 5 \ \text{inches with an area of} \ 25 \ \text{sq in.}

Example C

Rectangle NPQR is separated into 4 smaller rectangles.

Solution:

We can use problem solving steps to help.

& \mathbf{Describe:} && \text{The large rectangle contains} \ 4 \ \text{smaller rectangles. The areas of three of the}\\&&& \text{rectangles are given.} \ P \ \text{is a square. All dimensions are whole numbers.}\\\\& \mathbf{My \ Job:} && \text{Use the given areas. Figure out the area of Rectangle} \ R.\\\\& \mathbf{Plan:} && \text{Find common factors of the areas.} \ P \ \text{is a square, so the dimensions can be easily}\\&&& \text{determined. Figure out the dimensions of Rectangles} \ N \ \text{and} \ Q. \ \text{This will give the}\\&&& \text{dimensions for Rectangle} \ R. \ \text{Use the area formula to figure out the area of}\\&&& \text{Rectangle} \ R.\\\\& \mathbf{Solve:} && P \ \text{is a square, so the dimensions are} \ 6 \ \text{in. by} \ 6 \ \text{in. That means that one of the}\\&&& \text{dimensions of Rectangle} \ N \ \text{is} \ 6 \ \text{inches. The area is} \ 48 \ \text{square inches, so the other}\\&&& \text{dimension must be} \ 48 \div 6, \ \text{or} \ 8 \ \text{inches. Likewise, one of the dimensions of}\\&&& \text{Rectangle} \ Q \ \text{is} \ 6, \ \text{so the other is} \ 24 \div 6, \ \text{or} \ 4 \ \text{inches. The dimensions of Rectangle}\\&&& R \ \text{are} \ 8 \ \text{inches from the shared side with Rectangle} \ N \ \text{and} \  4 \ \text{inches form the}\\&&& \text{shared side with Rectangle} \ Q. \ \text{The area of} \ R \ \text{is} \ 8 \times 4, \ \text{or} \ 32 \ \text{square inches.}\\\\& \mathbf{Check:} && N: 8 \ \text{by} \ 6 \ \text{inches with an area of} \ 48 \ \text{sq in.}\\&&& P: 6 \ \text{by} \ 6 \ \text{inches with an area of} \ 36 \ \text{sq in.}\\&&& Q: 6 \ \text{by} \ 4 \ \text{inches with an area of} \ 24 \ \text{sq in.}\\&&& R: 8 \ \text{by} \ 4 \ \text{inches with an area of} \ 32 \ \text{sq in.}

Concept Problem Revisited

We can use problem solving steps to help.

& \mathbf{Describe:} && \text{The large rectangle contains} \ 4 \ \text{smaller rectangles. The areas of three of the}\\&&& \text{rectangles are given.} \ A \ \text{is a square. All dimensions are whole numbers.}\\\\& \mathbf{My \ Job:} && \text{Use the given areas. Figure out the area of Rectangle} \ C.\\\\& \mathbf{Plan:} && \text{Find common factors of the areas.} \ A \ \text{is a square, so the dimensions can be easily}\\&&& \text{determined. Figure out the dimensions of Rectangles} \ B \ \text{and} \ D. \ \text{This will give the}\\&&& \text{dimensions for Rectangle} \ C. \ \text{Use the area formula to figure out the area of}\\&&& \text{Rectangle} \ C.\\\\& \mathbf{Solve:} && A \ \text{is a square, so the dimensions are} \ 4 \ \text{in. by} \ 4 \ \text{in. That means that one of the}\\&&& \text{dimensions of Rectangle} \ B \ \text{is} \ 4 \ \text{inches. The area is} \ 24 \ \text{square inches, so the other}\\&&& \text{dimension must be} \ 24 \div 4, \ \text{or} \ 6 \ \text{inches. Likewise, one of the dimensions of}\\&&& \text{Rectangle} \ D \ \text{is} \ 4, \ \text{so the other is} \ 20 \div 4, \ \text{or} \ 5 \ \text{inches. The dimensions of Rectangle}\\&&& C \ \text{are} \ 6 \ \text{inches from the shared side with Rectangle} \ B \ \text{and} \ 5 \ \text{inches form the}\\&&& \text{shared side with Rectangle} \ D. \ \text{The area of} \ C \ \text{is} \ 6 \times 5, \ \text{or} \ 30 \ \text{square inches.}\\\\& \mathbf{Check:} && A: 4 \ \text{by} \ 4 \ \text{inches with an area of} \ 16 \ \text{sq in.}\\&&& B: 4 \ \text{by} \ 6 \ \text{inches with an area of} \ 24 \ \text{sq in.}\\&&& C: 5 \ \text{by} \ 5 \ \text{inches with an area of} \ 30 \ \text{sq in.}\\&&& D: 4 \ \text{by} \ 5 \ \text{inches with an area of} \ 20 \ \text{sq in.}

Vocabulary

A rectangle is a 4-sided shape whose angles are all right angles. The dimensions of a rectangle are the lengths of its sides (usually called the length and width ). In general, area is a calculation of the number of unit squares it takes to fill up a shape.

Guided Practice

1. Rectangle EFGH is separated into 4 smaller rectangles.

2. Rectangle JKLM is separated into 4 smaller rectangles.

3. Rectangle NPQR is separated into 4 smaller rectangles.

Answers:

1. 6 square inches

2. 56 square inches

3. 8 square inches

Practice

For each problem below, the large rectangle has been separated into 4 smaller rectangles.

Image Attributions

Description

Difficulty Level:

At Grade

Tags:

Grades:

Date Created:

Jan 18, 2013

Last Modified:

May 27, 2014
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