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2.3 Quadrilaterals

Types Properties of Quadrilaterals • Perimeter Area

A quadrilateral is a closed plane figure with four sides, these four sides form four interior angles. A general quadrilateral is shown in Fig. 2.51.

A quadrilateral or 4 sided polygon that is closed and has 4 interior angles. — Fig. 2.51

A diagonal of a polygon is a straight line segment joining any two nonadjacent vertices. The dashed line is one of two diagonals of the quadrilateral in Fig. 2.52.

A quadrilateral with a diagonal from one vertex to another, non-adjacent vertex. — Fig. 2.52

TYPES OF QUADRILATERALS

A parallelogram is a quadrilateral in which opposite sides are parallel. In a parallelogram, opposite sides are equal opposite angles are equal. A rhombus is a parallelogram with four equal sides.

A rectangle is a parallelogram in which intersecting sides are perpendicular, which means that all four interior angles are right angles. In a rectangle, the longer side is usually called the length, the shorter side is called the width. A square is a rectangle with four equal sides.

A trapezoid is a quadrilateral in which two sides are parallel. The parallel sides are called the bases of the trapezoid.

EXAMPLE 1 Types of quadrilaterals

A parallelogram is shown in Fig. 2.53(a). Opposite sides a are equal in length, as are opposite sides b. A rhombus with equal sides s is shown in Fig. 2.53(b). A rectangle is shown in Fig. 2.53(c). The length is labeled l, the width is labeled w. A square with equal sides s is shown in Fig. 2.53(d). A trapezoid with bases $b_{1}$ $b_{1}$ $b_{2}$ $b_{2}$ is shown in Fig. 2.53(e).

PERIMETER AREA OF A QUADRILATERAL

The perimeter of a quadrilateral is the sum of the lengths of the four sides.

EXAMPLE 2 Perimeter—window molding

An architect designs a room with a rectangular window 36 in. high 21 in. wide, with another window above in the shape of an equilateral triangle, 21 in. on a side. See Fig. 2.54. How much molding is needed for these windows?

A rectangular window has length 36 inches and width 21 inches. A triangular window has 3 sides of 21 inches. — Fig. 2.54

The length of molding is the sum of the perimeters of the windows. For the rectangular window, the opposite sides are equal, which means the perimeter is twice the length l plus twice the width w. For the equilateral triangle, the perimeter is three times the side s. Therefore, the length L of molding is

\begin{aligned} L & = & 2 l + 2 w + 3 s \\ = & 2 (36) + 2 (21) + 3 (21) \\ = & 177 in . \end{aligned}

$\begin{aligned} L & = & 2 l + 2 w + 3 s \\ = & 2 (36) + 2 (21) + 3 (21) \\ = & 177 in . \end{aligned}$

We could write down formulas for the perimeters of the different kinds of triangles quadrilaterals. However, if we remember the meaning of perimeter as being the total distance around a geometric figure, such formulas are not necessary.

For the areas of the square, rectangle, parallelogram, trapezoid, we have the following formulas.

A = s^{2} Square of side s (Fig . 2.55)

$A = s^{2} Square of side s (Fig . 2.55)$ (2.5)

A = l w Rectangle of length l width w (Fig . 2.56)

$A = l w Rectangle of length l width w (Fig . 2.56)$ (2.6)

A = b h Parallelogram of base b height h (Fig . 2.57)

$A = b h Parallelogram of base b height h (Fig . 2.57)$ (2.7)

A = \frac{1}{2} h (b_{1} + b_{2}) Trapezoid of bases b_{1} b_{2} height h (Fig . 2.58)

$A = \frac{1}{2} h (b_{1} + b_{2}) Trapezoid of bases b_{1} b_{2} height h (Fig . 2.58)$ (2.8)

A square with sides measuring s. — Fig. 2.55

A rectangle with length l and width w. — Fig. 2.56

A parallelogram with base b and height h. — Fig. 2.57

A trapezoid with bases b 1 and b 2, and height h. — Fig. 2.58

Because a rectangle, a square, a rhombus are special types of parallelograms, the area of these figures can be found from Eq. (2.7). The area of a trapezoid is of importance when we find areas of irregular geometric figures in Section 2.5.

EXAMPLE 3 Area—park design

A city park is designed with lawn areas in the shape of a right triangle, a parallelogram, a trapezoid, as shown in Fig. 2.59, with walkways between them. Find the area of each section of lawn the total lawn area.

Three areas. A right triangle has legs of 45 feet and 72 feet. A parallelogram has a side of 72 feet and height of 45 feet. A trapezoid has bases of 35 feet and 72 feet, and a height of 45 feet. — Fig. 2.59

\begin{aligned} A_{1} & = & \frac{1}{2} b h = \frac{1}{2} (72) (45) = 1600 {ft}^{2} \\ A_{2} & = & b h = (72) (45) = 3200 {ft}^{2} \\ A_{3} & = & \frac{1}{2} h (b_{1} + b_{2}) = \frac{1}{2} (45) (72 + 35) = 2400 {ft}^{2} \end{aligned}

$\begin{aligned} A_{1} & = & \frac{1}{2} b h = \frac{1}{2} (72) (45) = 1600 {ft}^{2} \\ A_{2} & = & b h = (72) (45) = 3200 {ft}^{2} \\ A_{3} & = & \frac{1}{2} h (b_{1} + b_{2}) = \frac{1}{2} (45) (72 + 35) = 2400 {ft}^{2} \end{aligned}$

The total lawn area is about ${7200 ft}^{2} .$ ${7200 ft}^{2} .$

EXAMPLE 4 Perimeter—computer chip dimensions

The length of a rectangular computer chip is 2.0 mm longer than its width. Find the dimensions of the chip if its perimeter is 26.4 mm.

Because the dimensions, the length the width, are required, let $w = the$ $w = the$ width of the chip. Because the length is 2.0 mm more than the width, we know that $w + 2.0 = the$ $w + 2.0 = the$ length of the chip. See Fig. 2.60.

A rectangle with length w plus 2.0 and width w. — Fig. 2.60

Because the perimeter of a rectangle is twice the length plus twice the width, we have the equation

2 (w + 2.0) + 2 w = 26.4

$2 (w + 2.0) + 2 w = 26.4$

because the perimeter is given as 26.4 mm. This is the equation we need.

Solving this equation, we have

\begin{aligned} 2 w + 4.0 + 2 w & = & 26.4 \\ 4 w & = & 22.4 \\ w & = & 5.6 mm w + 2.0 = 7.6 mm \end{aligned}

$\begin{aligned} 2 w + 4.0 + 2 w & = & 26.4 \\ 4 w & = & 22.4 \\ w & = & 5.6 mm w + 2.0 = 7.6 mm \end{aligned}$

Therefore, the length is 7.6 mm the width is 5.6 mm. These values check with the statements of the original problem.

EXERCISES 2.3

In Exercises 1–4, make the given changes in the indicated examples of this section then solve the given problems.

In Example 1, interchange the lengths of $b_{1}$ $b_{1}$ $b_{2}$ $b_{2}$ in Fig. 2.53(e). What type of quadrilateral is the resulting figure?
In Example 2, change the equilateral triangle of side 21 in. to a square of side 21 in. then find the length of molding.
In Example 3, change the dimension of 45 ft to 55 ft in each figure then find the area.
In Example 4, change 2.0 mm to 3.0 mm, then solve.

In Exercises 5–12, find the perimeter of each figure.

Square: side of 85 m
Rhombus: side of 2.46 ft
Rectangle: $l = 9.200 in ., w = 7.420 in .$ $l = 9.200 in ., w = 7.420 in .$
Rectangle: $l = 142 cm, w = 126 cm$ $l = 142 cm, w = 126 cm$
Parallelogram in Fig. 2.61

Fig. 2.61
Parallelogram in Fig. 2.62

Fig. 2.62
Trapezoid in Fig. 2.63

Fig. 2.63
Trapezoid in Fig. 2.64

Fig. 2.64

In Exercises 13–20, find the area of each figure.

Square: $s = 6.4 mm$ $s = 6.4 mm$
Square: $s = 15.6 ft$ $s = 15.6 ft$
Rectangle: $l = 8.35 in ., w = 2.81 in .$ $l = 8.35 in ., w = 2.81 in .$
Rectangle: $l = 142 cm, w = 126 cm$ $l = 142 cm, w = 126 cm$
Parallelogram in Fig. 2.61
Parallelogram in Fig. 2.62
Trapezoid in Fig. 2.63
Trapezoid in Fig. 2.64

In Exercises 21–24, set up a formula for the indicated perimeter or area. (Do not include dashed lines.)

The perimeter of the figure in Fig. 2.65 (a parallelogram a square attached)

Fig. 2.65
The perimeter of the figure in Fig. 2.66 (two trapezoids attached)

Fig. 2.66
Area of figure in Fig. 2.65
Area of figure in Fig. 2.66

In Exercises 25–46, solve the given problems.

If the angle between adjacent sides of a parallelogram is $90 °,$ $90 °,$ what conclusion can you make about the parallelogram?
What conclusion can you make about the two triangles formed by the sides diagonal of a parallelogram? Explain.
Find the area of a square whose diagonal is 24.0 cm.
In a trapezoid, find the angle between the bisectors of the two angles formed by the bases one nonparallel side.
Noting the quadrilateral in Fig. 2.67, determine the sum of the interior angles of a quadrilateral.

Fig. 2.67
The sum S of the measures of the interior angles of a polygon with n sides is $S = 180 (n - 2) .$ $S = 180 (n - 2) .$ (a) Solve for n. (b) If $S = 3600 °,$ $S = 3600 °,$ how many sides does the polygon have?
Express the area A of the large rectangle in Fig. 2.68 formed by the smaller rectangles in two ways. What property of numbers is illustrated by the results?

Fig. 2.68
Express the area of the square in Fig. 2.69 in terms of the smaller rectangles into which it is divided. What algebraic expression is illustrated by the results?

Fig. 2.69

Figure 2.69 Full Alternative Text
Noting how a diagonal of a rhombus divides an interior angle, explain why the automobile jack in Fig. 2.70 is in the shape of a rhombus.

Fig. 2.70
Part of an electric circuit is wired in the configuration of a rhombus one of its altitudes as shown in Fig. 2.71. What is the length of wire in this part of the circuit?

Fig. 2.71
A walkway 3.0 m wide is constructed along the outside edge of a square courtyard. If the perimeter of the courtyard is 320 m, (a) what is the perimeter of the square formed by the outer edge of the walkway? (b) What is the area of the walkway?
An architect designs a rectangular window such that the width of the window is 18 in. less than the height. If the perimeter of the window is 180 in., what are its dimensions?
Find the area of the cross section of concrete highway support shown in Fig. 2.72. All measurements are in feet are exact.

Fig. 2.72

Figure 2.72 Full Alternative Text
A beam support in a building is in the shape of a parallelogram, as shown in Fig. 2.73. Find the area of the side of the beam shown.

Fig. 2.73
Each of two walls (with rectangular windows) of an A-frame house has the shape of a trapezoid as shown in Fig. 2.74. If a gallon of paint covers ${320 ft}^{2},$ ${320 ft}^{2},$ how much paint is required to paint these walls? (All data are accurate to two significant digits.)

Fig. 2.74
A 1080p high-definition widescreen television screen has 1080 pixels in the vertical direction 1920 pixels in the horizontal direction. If the screen measures 15.8 in. high 28.0 in. wide, find the number of pixels per square inch.
The ratio of the width to the height of a 43.3 cm (diagonal) laptop computer screen is 1.60. What is the width w height h of the screen?
Six equal trapezoidal sections form a conference table in the shape of a hexagon, with a hexagonal opening in the middle. See Fig. 2.75. From the dimensions shown, find the area of the table top.

Fig. 2.75
A fenced section of a ranch is in the shape of a quadrilateral whose sides are 1.74 km, 1.46 km, 2.27 km, 1.86 km, the last two sides being perpendicular to each other. Find the area of the section.
A rectangular security area is enclosed on one side by a wall, the other sides are fenced. The length of the wall is twice the width of the area. The total cost of building the wall fence is $13,200. If the wall costs $50.00/m the fence costs $5.00/m, find the dimensions of the area.
What is the sum of the measures of the interior angles of a quadrilateral? Explain.
Find a formula for the area of a rhombus in terms of its diagonals $d_{1}$ $d_{1}$ $d_{2} .$ $d_{2} .$ (See Exercise 33.)

Answers to Practice Exercises

$d = \sqrt{l^{2} + w^{2}}$ $d = \sqrt{l^{2} + w^{2}}$
$p = 4 s$ $p = 4 s$
${43 mm}^{2}$ ${43 mm}^{2}$

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