Chapter 8: Q. 16 (page 556)

Given that $∆ O A B$ is an isosceles triangle and the shaded sector is a semicircle, find the area of the entire region. Express your answer as a decimal rounded to two places.

Short Answer

Expert verified

The area is $17.22$ square units.

Step by step solution

Step 1. Given Information

The given figure is

Step 2. Find the Angles of Triangle

The triangle is an isosceles triangle.
Sum of the angles of a triangle is $180 °$ .
So,

$\begin{array}{rcl} 40 ° + x + x & = & 180 ° \\ 2 x & = & 140 ° \\ x & = & 70 ° \end{array}$

Step 3. Use rule of Sines

Use the rule of sines to find the missing side, which is also the diameter of the semicircle.

$\begin{array}{rcl} \frac{\sin 40}{a} & = & \frac{\sin 70}{5} \\ a & = & 5 \frac{\sin 40}{\sin 70} \\ = & 5 \times 0.68 \\ = & 3.42 \end{array}$

Step 4. Find the total area

The total area of the figure is the sum of the area of the triangle and the semicircle.
To find the area of the triangle, find the semiperimeter.

$s = \frac{3.42 + 5 + 5}{2} = 6.71$

So, the area of the triangle is calculated below.

$\begin{array}{rcl} A_{T} & = & \sqrt{6.71 \times 1.71 \times 1.71 \times 3.29} \\ = & 8.03 \end{array}$

And the area of the semicircle is calculated as shown below:

$A_{C} = π {(\frac{3.42}{2})}^{2} = 9.19$

So, the total area is $8.03 + 9.19 = 17.22$ square units.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Given that $∆ O A B$ is an isosceles triangle and the shaded sector is a semicircle, find the area of the entire region. Express your answer as a decimal rounded to two places.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Find the Angles of Triangle

Step 3. Use rule of Sines

Step 4. Find the total area

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Mechanics Maths

Decision Maths

Calculus

Logic and Functions

Probability and Statistics

Study anywhere. Anytime. Across all devices.

Given that ∆OABis an isosceles triangle and the shaded sector is a semicircle, find the area of the entire region. Express your answer as a decimal rounded to two places.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Find the Angles of Triangle

Step 3. Use rule of Sines

Step 4. Find the total area

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Mechanics Maths

Decision Maths

Calculus

Logic and Functions

Probability and Statistics

Study anywhere. Anytime. Across all devices.

Given that $∆ O A B$ is an isosceles triangle and the shaded sector is a semicircle, find the area of the entire region. Express your answer as a decimal rounded to two places.