My Math Forum  

Go Back   My Math Forum > High School Math Forum > Geometry

Geometry Geometry Math Forum


Thanks Tree2Thanks
  • 1 Post By skipjack
  • 1 Post By Yooklid
Reply
 
LinkBack Thread Tools Display Modes
March 29th, 2017, 11:12 AM   #1
Newbie
 
Joined: Mar 2017
From: Netherlands

Posts: 1
Thanks: 0

Line length from mid-point circle to outside diameter circle

I might be stupid, but I have been looking at this for a few hours: either I am stupid or it is unsolvable (which could be an option).
I have two circles, one with a diameter of 16mm, the other one with diameter of 32mm. A line which starts at the mid-point of the 32mm circle, connects to the outside of the 16mm circle. The angle it makes is 20 degrees. How should I calculate the length of this line? I tried a lot of triangle equations but I did not manage to get the answer... I do not need the answer, I need the calculation how its done actually. If someone knows how to do it, please tell me how I should do it, you do not have to write everything down if you do not want to.



Thanks!
Byte is offline  
 
March 29th, 2017, 03:06 PM   #2
Global Moderator
 
Joined: Dec 2006

Posts: 18,166
Thanks: 1424

The greatest value angle a can have is approximately 19.47122°, so your configuration is impossible.
Thanks from topsquark
skipjack is offline  
March 29th, 2017, 03:26 PM   #3
Senior Member
 
Joined: Jul 2008
From: Western Canada

Posts: 3,270
Thanks: 38

Well, first of all there's a problem with your diagram. A line from the centre of the larger circle that's tangent to the smaller one makes an angle of only 19.47° with the line between centres. So, your angle of 20° is impossible. The line would never touch the smaller circle.

Oops. Skipjack beat me to it.
Thanks from topsquark
Yooklid is offline  
March 29th, 2017, 04:56 PM   #4
Senior Member
 
Joined: Jul 2008
From: Western Canada

Posts: 3,270
Thanks: 38

Here is a general solution (refer to attached diagram) when the angle a is small enough for a valid solution.



The things we know are shown in green.

We can first solve for the unknown angles b and c using the law of sines:

$\dfrac{\sin(a)}{A}=\dfrac{\sin(b)}{B}=\dfrac{\sin (c)}{C}$

Since we know the lengths A and B and angle a, we can use this formula to solve for angle b:

$\dfrac{\sin(a)}{A}=\dfrac{\sin(b)}{B}$

Rearranging:

$b=\arcsin(\frac{B}{A}\sin(a))$

We have to be careful here though. The arcsin function has two possible results, an acute angle and an obtuse angle. This function will normally return the smaller value. We can see from the diagram, that angle b is obtuse, and so we have to subtract the arcsin value from 180°, thus:

$b=180-\arcsin(\frac{B}{A}\sin(a))$

Now that we know angles a and b, we can use the fact that the included angles of a triangle add up to 180°. Therefore:
$c=180-b-a$

We now know all of the angles and two sides A and B. We can now apply the law of cosines to find the unknown side C. The law of cosines is:

$C^2=A^2+B^2-2AB\cos(c)$

Therefore:

$C=\sqrt{A^2+B^2-2AB\cos(c)}$
.
Attached Images
File Type: jpg Geometry01.jpg (5.9 KB, 12 views)

Last edited by Yooklid; March 29th, 2017 at 05:00 PM.
Yooklid is offline  
Reply

  My Math Forum > High School Math Forum > Geometry

Tags
circle, diameter, length, line, midpoint



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Length of a drawn circle mark eaton24 Applied Math 15 January 19th, 2016 04:09 AM
Minimum diameter of a circle to fit other circles inside Ash Algebra 1 March 12th, 2014 10:50 AM
finding a point on a circle given the slope of the tan line mathkid Calculus 6 October 1st, 2012 07:49 PM
Finding The Standard Equation Of A Circle Given a Point/Line soulrain Algebra 7 January 5th, 2012 08:51 PM
the point through a line closest to the unit circle mad Algebra 3 August 18th, 2010 01:20 PM





Copyright © 2017 My Math Forum. All rights reserved.