Chapter 8: Q. 77 (page 519)

Geometry Find the value of the angle $θ$ in degrees rounded to the nearest tenth of a degree.

Short Answer

Expert verified

The answer is $θ = 38.9 ° .$

Step by step solution

Step 1. Given information

We have two circles small circle have radius r1 = 2 and bigger circle have radius r2 = 4.

These two circle has two common tangents meeting at some point at an angle $θ$ .

Given:

$P S = 2 Q C = 4 ∠ P O S = ∠ Q O C = \frac{θ}{2} . (S i n c e i t s y m m e t r i c a b o u t t h e l i n e d r a w n f r o m O) . S C = 2 + 4 = 6, A n d ∠ O P S = ∠ O Q C = 90 ° . (S i n c e t h e \tan g e n t i s p e r p e n d i c u l a r t o t h e c i r c l e .)$

Step 2. Triangles and its properties

$I n ∆ P O S, \sin (\frac{θ}{2}) = \frac{P S}{O S} . I n ∆ Q O C, \sin (\frac{θ}{2}) = \frac{Q C}{O C} = \frac{Q C}{O S + S C} .$

Step 3. Solving the equations from step 2

$\sin (\frac{θ}{2}) = \frac{P S}{O S} = \frac{2}{O S}, N o w, \sin (\frac{θ}{2}) = \frac{Q C}{O S + S C}, \frac{2}{O S} = \frac{4}{O S + 6}, S o l v i n g f o r O S w e g e t, 2 O S + 12 = 4 O S, 2 O S = 12, O S = 6, N o w f r o m a b o v e e q u a t i o n, \sin (\frac{θ}{2}) = \frac{2}{O S} = \frac{2}{6} = \frac{1}{3}, \frac{θ}{2} = \sin^{- 1} (\frac{1}{3}), θ = 2 \sin^{- 1} (\frac{1}{3}), θ = 38.9424 . . . . ° θ ≅ 38.9 ° .$

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.

Geometry Find the value of the angle $θ$ in degrees rounded to the nearest tenth of a degree.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Triangles and its properties

Step 3. Solving the equations from step 2

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Pure Maths

Theoretical and Mathematical Physics

Logic and Functions

Applied Mathematics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Geometry Find the value of the angle θin degrees rounded to the nearest tenth of a degree.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Triangles and its properties

Step 3. Solving the equations from step 2

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Pure Maths

Theoretical and Mathematical Physics

Logic and Functions

Applied Mathematics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Geometry Find the value of the angle $θ$ in degrees rounded to the nearest tenth of a degree.