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Core Connections Geometry, 2013

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Core Connections Geometry, 2013 View details

2. Section 6.2

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Exercise 76 Page 376

Use trigonometric ratios to find the dimensions you need to calculate the trapezoid's perimeter and area.

Area= 100.5 yds^2
Perimeter= 43.35 yds

Practice makes perfect

We are supposed to find the area and perimeter of the trapezoid.

Area

To find the area, we have to know the length of the trapezoid's parallel sides and its height. If we mark the height in our diagram, we get a right triangle. In this triangle, we know the hypotenuse and we want to find the opposite leg of the reference angle.

Let's solve for h in this equation.

sin 65^(∘)=h/8

▼

Solve for h

LHS * 8=RHS* 8

8sin 65^(∘)=h

Rearrange equation

h=8sin 65^(∘)

Use a calculator

h=7.25046...

Round to 2 decimal place(s)

h≈ 7.25

Next, we need to find the measure of the second parallel side. For this purpose, we will divide it into three different segments. In two segments, the length is unknown. To determine these lengths, we will write trigonometric equations.

By solving for a and b we can find the length of the second parallel side by adding their lengths with the middle segment.

cos 65^(∘) =a/8

▼

Solve for a

LHS * 8=RHS* 8

8cos 65^(∘) = a

Rearrange equation

a =8cos 65^(∘)

Use a calculator

a = 3.38094...

Round to 2 decimal place(s)

a ≈ 3.38

Let's also solve for b.

tan 72^(∘) =7.25/b

▼

Solve for b

LHS * b=RHS* b

tan 72^(∘) * b = 7.25

.LHS /sin 72^(∘).=.RHS /sin 72^(∘).

b = 7.25/tan 72^(∘)

Calculate quotient

b = 2.35566...

Round to 2 decimal place(s)

b ≈ 2.36

With this, we can determine the length of the second parallel side. 3.38+11+2.36 = 16.73 yds Now we have all the dimensions we need to calculate the trapezoid's area.

Perimeter

To calculate the perimeter, we need to find the length of the remaining side which we will label c. To do that, we can use the sine ratio.

By solving for c we can find the length of the remaining side.

sin 72^(∘) =7.25/c

▼

Solve for c

LHS * c=RHS* c

sin 72^(∘) * c =7.25

.LHS /sin 72^(∘).=.RHS /sin 72^(∘).

c =7.25/sin 72^(∘)

Use a calculator

c =7.25/sin 72^(∘)

Calculate quotient

c =7.62310...

Round to 2 decimal place(s)

c ≈ 7.62

Now we can calculate the perimeter.

Subchapter links

6.2.1 Angles on a Pool Table p.368-369

6.2.2 Investigating a Triangle p.371-372

6.2.3 Creating a Mathematical Model p.375-376

6.2.4 Analyzing a Game p.379-380

6.2.5 Using Transformations and Symmetry to Design Snowflakes p.384-385