Chapter 8.4, Problem 56PPS

a.

To determine

To find: The area of each of the squares.

Answer to Problem 56PPS

area of the larger square is $a^2$ .

area of the smaller square is $b^2$

Explanation of Solution

Given information:

Each edge of the paper is $a$

If any corners, draw a smaller square and label the edges $b$ .

To find the area of a square find the square of the length of the sides.

The area of the larger square is $\left(a\right)\left(a\right)=a^2$ .

The area of the smaller square is $\left(b\right)\left(b\right)=b^2$

b.

To determine

To find: Area of the shape.

Answer to Problem 56PPS

Area is $a^2-b^2$

Explanation of Solution

Given information:

Each edge of the paper is $a$

If any corners, draw a smaller square and label the edges $b$ .

To find the area of the shape after cutting the smaller square out of the corner,subtract the area of the smaller square from the area of the larger square.

So, the area is $a^2-b^2$

c.

To determine

To find: The length of the new arrangement, width and area.

Answer to Problem 56PPS

Area of the new arrangement is $a^2-b^2$

Explanation of Solution

Given information:

Each edge of the paper is $a$

If any corners, draw a smaller square and label the edges $b$ .

The length of the new arrangement is $a+b$ .

The width of the new arrangement is $a-b$

So, the area of the new arrangement is $\left(a+b\right)\left(a-b\right)=a^2-b^2$

d.

To determine

To find: What pattern does this verify.

Explanation of Solution

This pattern demonstrates the product of a sum and a difference,

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