Chapter 4.6, Problem 28WE

a)

To determine

To find: the sum of areas of four smaller rectangles.

Answer to Problem 28WE

The sum of areas of four smaller rectangles is $\mathrm{a}\mathrm{c}+\mathrm{a}\mathrm{d}+\mathrm{b}\mathrm{c}+\mathrm{b}\mathrm{d}$ .

Explanation of Solution

Given information:

The given figure is shown below.

  

Formula used:

Area of a rectangle with length x and breadth y

  $\text{Area}=\mathrm{x}\mathrm{y}$

Calculation:

Consider the given figure.

  

There are four small rectangles with length a and breadth c , length b and breadth c , length a and breadth d , and length b and breadth d .

Calculate area of the rectangle with length a , and breadth c .

  ${\text{A}}_1=\mathrm{a}\mathrm{c}$

Calculate area of the rectangle with length a , and breadth d .

  ${\text{A}}_2=\mathrm{a}\mathrm{d}$

Calculate area of the rectangle with length b , and breadth c .

  ${\text{A}}_3=\mathrm{b}\mathrm{c}$

Calculate area of the rectangle with length b , and breadth d .

  ${\text{A}}_4=\mathrm{b}\mathrm{d}$

Calculate sum of the areas of small rectangles as shown:

  ${\text{A}}_1+{\text{A}}_2+{\text{A}}_3+{\text{A}}_4=\mathrm{a}\mathrm{c}+\mathrm{a}\mathrm{d}+\mathrm{b}\mathrm{c}+\mathrm{b}\mathrm{d}$

b)

To determine

To find: the product of length and breadth of original rectangle.

Answer to Problem 28WE

The product of length and breadth of original rectangle is $\mathrm{a}\mathrm{c}+\mathrm{a}\mathrm{d}+\mathrm{b}\mathrm{c}+\mathrm{b}\mathrm{d}$ .

Explanation of Solution

Given information:

The given figure is shown below.

  

Formula used:

Area of a rectangle with length x and breadth y

  $\text{Area}=\mathrm{x}\mathrm{y}$

Calculation:

Consider the given figure.

  

There are four small rectangles with length a and breadth c , length b and breadth c , length a and breadth d , and length b and breadth d .

Length of the original rectangle is $\mathrm{a}+\mathrm{b}$ unit.

Breadth of the original rectangle is $\mathrm{c}+\mathrm{d}$ unit.

Calculate the product of length and breadth of original rectangle as shown:

  $\begin{array}{c}\left(\mathrm{a}+\mathrm{b}\right)\left(\mathrm{c}+\mathrm{d}\right)=\mathrm{a}\left(\mathrm{c}+\mathrm{d}\right)+\mathrm{b}\left(\mathrm{c}+\mathrm{d}\right)\\ =\mathrm{a}\mathrm{c}+\mathrm{a}\mathrm{d}+\mathrm{b}\mathrm{c}+\mathrm{b}\mathrm{d}\end{array}$

c)

To determine

To explain: the similarity in the answer of part (a) and part (b).

Answer to Problem 28WE

The answer of part (a) and part (b) is same.

Explanation of Solution

Given information:

The given figure is shown below.

  

Formula used:

Area of a rectangle with length x and breadth y

  $\text{Area}=\mathrm{x}\mathrm{y}$

Calculation:

Consider the given figure.

  

There are four small rectangles with length a and breadth c , length b and breadth c , length a and breadth d , and length b and breadth d .

Length of the original rectangle is $\mathrm{a}+\mathrm{b}$ unit.

Breadth of the original rectangle is $\mathrm{c}+\mathrm{d}$ unit.

Calculate the product of length and breadth of original rectangle as shown:

  $\begin{array}{c}\left(\mathrm{a}+\mathrm{b}\right)\left(\mathrm{c}+\mathrm{d}\right)=\mathrm{a}\left(\mathrm{c}+\mathrm{d}\right)+\mathrm{b}\left(\mathrm{c}+\mathrm{d}\right)\\ =\mathrm{a}\mathrm{c}+\mathrm{a}\mathrm{d}+\mathrm{b}\mathrm{c}+\mathrm{b}\mathrm{d}\end{array}$

Calculate sum of the areas of small rectangles as shown:

  ${\text{A}}_1+{\text{A}}_2+{\text{A}}_3+{\text{A}}_4=\mathrm{a}\mathrm{c}+\mathrm{a}\mathrm{d}+\mathrm{b}\mathrm{c}+\mathrm{b}\mathrm{d}$

Area is same in both cases.

Therefore, area of original rectangle is equal to the sum of areas of small rectangles.