My Math Forum  

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

Algebra Pre-Algebra and Basic Algebra Math Forum


Thanks Tree2Thanks
Reply
 
LinkBack Thread Tools Display Modes
June 25th, 2011, 11:53 PM   #11
Senior Member
 
Joined: Apr 2007

Posts: 2,140
Thanks: 0

Let be the length from the vertex of the smaller base to the tangent point of the smaller circle.
Draw a line that passes through the centres of two circles. Now shift it units until the end of the line touches the vertex of smaller base.
Use Pythagorean theorem to determine the height
Since the line is shifted from where it crosses the two centres, height equals to the sum of two diameters.
Hence the height
Smaller base is
Hence to determine the larger base use the fact that larger base:smaller base = 9:1 by using the
tangent line of externally tangent circles as mid-base, two trapezoids are similar by 1:3.
Smaller base:mid-base = 1:3 and mid-base:larger base = 1:3 = 3:9 therefore smaller base:larger base = 1:9.
To determine the larger base, multiply the smaller base by 9 to get
johnny is offline  
 
June 26th, 2011, 09:00 AM   #12
Newbie
 
Joined: Jun 2011

Posts: 21
Thanks: 0

Re: Finding Sides and area of trapezoid around two circles.

I thought the height was just 8?
Spaghett is offline  
June 26th, 2011, 01:09 PM   #13
Senior Member
 
MarkFL's Avatar
 
Joined: Jul 2010
From: St. Augustine, FL., U.S.A.'s oldest city

Posts: 12,211
Thanks: 522

Math Focus: Calculus/ODEs
Re: Finding Sides and area of trapezoid around two circles.

My inclination was to approach this problem with analytic geometry (and like johnny assume an isosceles trapezoid). I oriented the system such that the smallest side of the trapezoid lies on the y-axis and the side opposite lies on the line x = 8. The centers of the circles lie on the x-axis where the smaller circle is then given by (x - 1) + y = 1 and the larger circle is given by (x - 5) + y = 3.

Now, the top side of the trapezoid must be tangent to the two circles. Let be the point of tangency with the smaller circle and be the point of tangency with the larger circle. We have that the line segments:

and are parallel and both perpendicular to the top side of the trapezoid. Thus, we have:



Square through:



Recall and giving:





Take the positive root:



Now, substitute for :







Thus, the slope of the top side of the trapezoid is:



Recall this slope is perpendicular to the segment giving:





Recall though that:



Equating, we find:





Combine like terms:











Thus the slope m of the top side is:



Now, using the point-slope formula, we find the equation of the line representing the top side as:



Put into slope-intercept form, we have:



Thus:

The smaller base b is

The larger base B is =\frac{18}{\sqrt{3}}=6\sqrt{3}" />

The two sides s are

The area A is

Note that these results agree completely with the results given by johnny, although I did not bother to rationalize the denominators.
MarkFL is offline  
June 26th, 2011, 01:18 PM   #14
Senior Member
 
mrtwhs's Avatar
 
Joined: Feb 2010

Posts: 714
Thanks: 151

Re: Finding Sides and area of trapezoid around two circles.

Quote:
Originally Posted by Spaghett
I thought the height was just 8?
There are at least two orientations. The surrounding trapezoid might be an isosceles trapezoid with height = 8 or it might be a trapezoid having two consecutive angles that are right angles (as I drew earlier). In this case I think the height is . I think these are the two extreme cases and that as you roll the small circle around the large one you can different trapezoids with heights ranging from up to 8.
mrtwhs is offline  
June 29th, 2011, 11:52 PM   #15
Senior Member
 
MarkFL's Avatar
 
Joined: Jul 2010
From: St. Augustine, FL., U.S.A.'s oldest city

Posts: 12,211
Thanks: 522

Math Focus: Calculus/ODEs
Re: Finding Sides and area of trapezoid around two circles.

I thought it might be fun (at least I thought so at first until I spent a great deal of time chasing down silly mistakes! ) to generalize a bit, and let the smaller circle have radius r and the larger circle have radius kr where k ? 1.

As before, we will assume an isosceles trapezoid and orient the system such that the smallest side of the trapezoid lies on the y-axis and the side opposite lies on the line x = 2r(k + 1). The centers of the circles lie on the x-axis where the smaller circle is then given by (x - r) + y = r and the larger circle is given by (x - r(2 + k)) + y = (kr).

Now, the top side of the trapezoid must be tangent to the two circles. Let be the point of tangency with the smaller circle and be the point of tangency with the larger circle. We have that the line segments:

and are parallel and both perpendicular to the top side of the trapezoid. Thus, we have:



Square through:



Recall and giving:





Take the positive root:



Now, substitute for :







Thus, the slope of the top side of the trapezoid is:



Recall this slope is perpendicular to the segment giving:



Cross-multiplication yields:



Recall that so we have:



Expansion, simplification, and solving for yields:









Thus, the slope m of the top side is:



Now, using the point-slope formula, we find the equation of the line representing the top side as:



Put into slope-intercept form, we have:



Thus:

The smaller base b is

The larger base B is

The two sides s are

The area A is
MarkFL is offline  
June 30th, 2013, 11:09 AM   #16
Newbie
 
Joined: Jun 2013

Posts: 2
Thanks: 0

Re: Finding Sides and area of trapezoid around two circles.

Ive seen people say the smaller base is sqrt(3) / 3 by using the pythagorean theorem.

Can someone go into a little more depth and explain how to use the pythagorean theorem on this problem when you only know the height, and you are assuming its an isoceles trapezoid?

You only have the one side of a right triangle, thats the height of 8, so i'm highly confused how people leap to the square root of three over three.

Thanks,
Samaurant is offline  
June 30th, 2013, 11:13 AM   #17
Newbie
 
Joined: Jun 2013

Posts: 2
Thanks: 0

Re:

Quote:
Originally Posted by johnny
Let be the length from the vertex of the smaller base to the tangent point of the smaller circle.
Draw a line that passes through the centres of two circles. Now shift it units until the end of the line touches the vertex of smaller base.
Use Pythagorean theorem to determine the height
Since the line is shifted from where it crosses the two centres, height equals to the sum of two diameters.
Hence the height
Smaller base is
Hence to determine the larger base use the fact that larger base:smaller base = 9:1 by using the
tangent line of externally tangent circles as mid-base, two trapezoids are similar by 1:3.
Smaller base:mid-base = 1:3 and mid-base:larger base = 1:3 = 3:9 therefore smaller base:larger base = 1:9.
To determine the larger base, multiply the smaller base by 9 to get
I dont understand your reasoning on how the base equals sqrt{3} / 3. You said "using pythagorean theorem" But to this point you only have one side to work with. where does the squart root of 3 come from?
Samaurant is offline  
Reply

  My Math Forum > High School Math Forum > Algebra

Tags
area, circles, finding, sides, trapezoid



Search tags for this page
Click on a term to search for related topics.
Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Calculation of sides and area of right-angled trapezoid perwin Algebra 17 November 17th, 2012 06:06 PM
Finding altitude of a part of trapezoid with given area... lei Algebra 4 May 1st, 2012 04:57 AM
re finding the area of circles with a regular octagon ibros06 Algebra 3 June 13th, 2010 07:50 PM
Det. length of sides of polygon from Area and ratio of sides telltree Algebra 0 January 21st, 2010 01:51 PM
I need help with circles and sides! jord peck Algebra 1 December 11th, 2007 06:24 PM





Copyright © 2019 My Math Forum. All rights reserved.