Chapter 4.3, Problem 36WE

To determine

To Find: The perimeter and area of the below shaded region

  

Answer to Problem 36WE

The required Area $=11x^2$

The required Perimeter $=20x$

Explanation of Solution

Given information:

The diagram:

  

Concept Used:

Rule of exponents for product of powers states that to multiply two powers with same base, the exponents are added.

For example $a^m\cdot a^n=a^{m+n}$

Commutative property of multiplication states that the product of any numbers remains the same irrespective of the order in which we multiply.

For example:

  $2\cdot 5=5\cdot 2=10$

Associative property of multiplication states that the product of the any numbers remains the same regardless of the grouping of factors.

For example:

  $2\left(5\cdot 3\right)=\left(2\cdot 5\right)\cdot 3=30$

Area of a rectangle $=length\cdot width$

Perimeter of a rectangle $=2\left(length+width\right)$

Calculation:

Name the given diagram and join DE such that $DE\parallel FG$

  

Now there is two Rectangle ABCE and DEFG

From the diagram:

  $\begin{array}{l}AC=2x\\ CD=x\\ FG=3x\\ BG=3x\\ CE=CD+DE\\ \because DE=FG\\ \therefore CE=x+3x=4x\\ Now,\\ BG=BE+EG\\ 3x=2x+EG\\ EG=3x-2x\\ EG=x\end{array}$

So, for rectangle ABCE

Length $=CE=4x$

Width $=AC=2x$

Area of rectangle ABCE $=length\cdot width$

Substituting the known value

Area of a rectangle $=4x\cdot 2x$

Commutative and Associative property of multiplication.

  $=\left(4\cdot 2\right)\left(x\cdot x\right)$

Using rule of exponents for product of powers

  $\begin{array}{l}=8x^{1+1}\\ =8x^2\end{array}$

Similarly, for rectangle DEFG

Length $=DE=3x$

Width $=EG=x$

Area of rectangle DEFG $=length\cdot width$

Substituting the known value

Area of rectangle DEFG $=3x\cdot x$

Commutative and Associative property of multiplication.

  $=3\left(x\cdot x\right)$

Using rule of exponents for product of powers

  $\begin{array}{l}=3x^{1+1}\\ =3x^2\end{array}$

Area of the shaded region = Area of rectangle ABCE + Area of rectangle DEFG

Substituting the known value

Area of the shaded region

  $\begin{array}{l}=8x^2+3x^2\\ =11x^2\end{array}$

Perimeter of rectangle ABCE $=2\left(length+width\right)$

Substituting the known value

Perimeter of rectangle ABCE $=2\left(4x+2x\right)$

  $\begin{array}{l}=2\cdot 6x\\ =12x\end{array}$

Perimeter of rectangle DEFG $=2\left(length+width\right)$

Substituting the known value

Perimeter of rectangle ABCE $=2\left(4x+2x\right)$

  $\begin{array}{l}=2\left(3x+x\right)\\ =2\cdot 4x\\ =8x\end{array}$

Perimeter of the shaded region = Perimeter of rectangle ABCE + Perimeter of rectangle DEFG

Substituting the known value

Perimeter of the shaded region

  $\begin{array}{l}=12x+8x\\ =20x\end{array}$

Substituting the known value

  $\begin{array}{l}=2\left(3ab+2ab\right)\\ =2\cdot 5ab\\ =10ab\end{array}$