Chapter 11, Problem 3RP

a.

To determine

To calculate:Thearea of shaded region which contains larger rectangle ABCDand smaller rectangle PQRS .

Answer to Problem 3RP

Thearea of shaded region is $70$ .

Explanation of Solution

Given information:

In rectangle ABCD, Side AD = $13$ ,

Side CD = $7$ .

In rectangle PQRS, Side PQ = $3$ ,

Side QR = $7$ .

Formula used:

Area of rectangle: $A=l\times w$

l = length of rectangle

w= width of triangle

Calculation:

  

Area of larger rectangle:

  $\begin{array}{l}A(\square ABCD)=BC\times AB\\ A(\square ABCD)=13\times 7=91\end{array}$

Area of smaller rectangle:

  $\begin{array}{l}A(\square PQRS)=QR\times PQ\\ A(\square PQRS)=7\times 3=21\end{array}$

Area of shaded region = Area of larger rectangle - Area of smaller rectangle

Area of shaded region $=91-21=70$

b.

To determine

To find: The area of shaded region which is a trapezoid ABCD.

Answer to Problem 3RP

The area of shaded region is $24$ .

Explanation of Solution

Given information:

In trapezoid ABCD, Side AD = $4$ ,

Side AB = $7$ .

Side DC = $5$ .

Formula used:

Area of trapezoid: $A=\frac{1}{2}\times (b_1+b_2)\times h$

b = base of trapezoid

h = height of trapezoid

Calculation:

  

Area of shaded region = Area of trapezoid ABCD Area of trapezoid ABCD :

  $\begin{array}{l}A=\frac{1}{2}\times (AB+DC)\times AD\\ A=\frac{1}{2}\times (7+5)\times 4\\ A=24\end{array}$

Area of shaded region = $24$

c.

To determine

To find: The area of shaded region which is anequilateral triangle XYZ.

Answer to Problem 3RP

The area of shaded region is $16\sqrt{3}$ .

Explanation of Solution

Given information:

In equilateral triangleXYZ, Side XY = YZ = XZ = $8$ .

Formula used:

Area of equilateral triangle: $A=\frac{s^2}{4}\sqrt{3}$

s = side of equilateral triangle

Calculation:

  

Area of shaded region = Area of equilateral triangle XYZ.

Area of equilateral triangle XYZ :

s=XY = YZ = XZ = $8$

  $\begin{array}{l}A=\frac{s^2}{4}\sqrt{3}\\ A=\frac{8^2}{4}\sqrt{3}\\ A=\frac{64}{4}\sqrt{3}\\ A=16\sqrt{3}\end{array}$

Area of shaded region = $16\sqrt{3}$