Chapter 8.6, Problem 43PPS

a.

To determine

Todraw: A square with side $a$ and then draw another square inside it with side $b$ which shares one vertex with the larger square, then find the area of the two squares.

Answer to Problem 43PPS

The area of the larger square is $a^2$ and the area of the smaller square is $b^2$ .

Explanation of Solution

Given:

Two squares of side $a$ and $b$ . The square with side $b$ is smaller and drawn inside the square with side $a$ .

Draw:

The two squares can be drawn as shown below.

  

Since the area of a square is given by the square of its side, the area of the larger square is $a^2$ and the area of the smaller square is $b^2$ .

b.

To determine

To find: The area of the remaining region obtained by cutting and removing the smaller square.

Answer to Problem 43PPS

  $a^2-b^2$

Explanation of Solution

Given:

Two squares of side $a$ and $b$ . The square with side $b$ is smaller and drawn inside the square with side $a$ .

Calculation:

The remaining region is shown below.

  

Now, the area of this region can be found by subtracting the area of the smaller square from the larger square.

Thus, the are of the remaining region is $a^2-b^2$ .

c.

To determine

To find: The dimension of the rectangle formed by rearranging the pieces cut from the remaining region through the diagonal joining the inside corner and the other corner.

Answer to Problem 43PPS

Length: $a+b$

Breadth: $a-b$

Explanation of Solution

Given:

Two squares of side $a$ and $b$ . The square with side $b$ is smaller and drawn inside the square with side $a$ .

Calculation:

The figure for the given situation can be drawn as shown below.

  

From the figure it can observed that the length of the rectangle is $a+b$ , and the breadth of the rectangle is $a-b$ .

d.

To determine

To write: The area of the rectangle as a product of two binomials.

Answer to Problem 43PPS

  $\left(a+b\right)\left(a-b\right)$

Explanation of Solution

From part (c), the length of the rectangle is $a+b$ , and the breadth of the rectangle is $a-b$ .

The area of a rectangle is given as the product of its length and breadth.

That is,

  $\text{Area of the rectangle = }\left(a+b\right)\left(a-b\right)$

e.

To determine

To complete: The given statement and state why it is true.

Answer to Problem 43PPS

  $a^2-b^2=\left(a+b\right)\left(a-b\right)$

Explanation of Solution

Given:

The statement: $a^2-b^2=\mathrm{......}$

Calculation:

From part (b), the area of the remaining region is $a^2-b^2$ . From part (d), area of the remaining region is $\left(a+b\right)\left(a-b\right)$ .

So, $a^2-b^2=\left(a+b\right)\left(a-b\right)$ .

The statement is true as it is an identity for difference of squares and represents the area of the remaining region.