Math Is Fun Forum

math137

Member

Registered: 2014-08-07

Posts: 12

Circle Geo Problems

Here's a few problems I need help with:

1.We are given a regular heptagon of side length 1. Let S be the set of points that are within a distance of 1 from some point on or inside the heptagon, but not including the heptagon itself. Find the area of S.

2. Let C and D be points on the semicircle with diameter \overline{AB}. If AD = BC = 3 and CD = 7, then find the radius of the semicircle.

Thx!

math137

Member

Registered: 2014-08-07

Posts: 12

Re: Circle Geo Problems

I have one more problem I need help with. This isn't really circle geo but...

3. Equilateral triangle ABC has centroid G. Triangle A'B'C' is the image of triangle ABC upon a dilation with center G and scale factor -2/3. Let K be the area of the region that is within both triangles. Find K/[ABC].

math137

Member

Registered: 2014-08-07

Posts: 12

Re: Circle Geo Problems

Could someone plz help? I really need help because it's due soon.

ElainaVW

Member

Registered: 2013-04-29

Posts: 580

Re: Circle Geo Problems

#1) Why does your drawing have rounded points like that?

Last edited by ElainaVW (2014-08-08 10:07:51)

math137

Member

Registered: 2014-08-07

Posts: 12

Re: Circle Geo Problems

I think it's supposed to. The heptagon is the white figure and then the set of points that are within a distance of 1 from some point on or inside the heptagon is the shaded part.

math137

Member

Registered: 2014-08-07

Posts: 12

Re: Circle Geo Problems

I actually got the first 2 problems. Thx EVW on your help on problem 2!

I still need help on this problem though: Equilateral triangle ABC has centroid G. Triangle A'B'C' is the image of triangle ABC upon a dilation with center G and scale factor -2/3. Let K be the area of the region that is within both triangles. Find K/[ABC]. I thought K was the area of the dilated equilateral triangle but that doesn't seem to be the case (the answer I got was 4/9).

Thanks!

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: Circle Geo Problems

For some strange reason I cannot post my help here.  EDIT: This problem has been solved by bobbym.  See post 9.

4/9 is right for the smaller triangle.  But you want the overlap.

I have done this before but I cannot find the thread so I'll have to start again.  ......................................

Right.  Here we go:

Let's start with setting AH = 9a.

As the centroid is 1/3 of the way up the median you can now work out in terms of a:

GH, then AG.

Lengths in A'B'C' will be 2/3 of the lengths in ABC so you can then work out in terms of a:

A'G and hence A'H

You can also work out A'J, B'L and hence JL.

You now know the relative sizes of ABC, CJL and A'JK.

So by adding and subtracting the right amounts you can work out the overlap in terms of ABC.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Circle Geo Problems

Hi Bob;

403 error?

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: Circle Geo Problems

Q1.

This diagram answers ElainaVW;s question.

To get points that are 1 unit from the straight sided heptagon you need straight lines completing the red bits, and parts of a circle radius 1 to go round the points.

It helps to be familiar with UK coins here.  The 50p and 20p coins have this property.  They have constant width and so can be correctly detected in slot machines.

So calculate 7 red bits, and 7 sectors.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Circle Geo Problems

This is a problem that up to now has only affected Agnishom and I. The server does not like some word combination that you have in your post. This is localized to a single thread usually. Do not try to force the post to pass, the server will then bar you. Rather, change the wording of the post until the error disappears.

I have entered the post for you.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: Circle Geo Problems

I'v also had that problem with the word "p.o.k.e.r".

---

“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: Circle Geo Problems

Exactly that did happen.  I tried to post it last night and got the error.  I tried again, then logged off and tried to get back on.  It wouldn't let me.  (ha!)  Then it got worse.  I couldn't even get the site up.  Had to email Rod.

Thanks for your help. 

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Circle Geo Problems

Hi;

I have entered your work in post #9. Please read post #10 for what to do to prevent being locked out or barred.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: Circle Geo Problems

Thanks for the post insert.  I've read and inwardly digested your advice.  yum

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

RandomPieKevin

Member

Registered: 2015-07-02

Posts: 29

Re: Circle Geo Problems

What did you guys get for Q3?

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: Circle Geo Problems

hi RandomPieKevin

I've had to re-work this.  Hope I've got it right.    I'm getting 11/27

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

RandomPieKevin

Member

Registered: 2015-07-02

Posts: 29

Re: Circle Geo Problems

Can you explain how you got it? Thanks.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: Circle Geo Problems

I'm using the diagram in post 7.  I need to label one more point.   H' is the point where AA' and C'B' cross.

As we only want ratios of areas it doesn't matter exactly what units I use, so I've simplified my working by assuming the easiest units.

Let's say the area of ABC is 81 units.

Then (using the scale factor 2/3) the area of A'B'C' must be 81 x2/3 x2/3 = 36

And let's say that AH = 9 units.

Then AG = 6 units, GH = 3 units.

(G is the centroid of both triangles, so I don't need G')

A'H' = 2/3 of 9 = 6

So A'G = 4  (and GH' = 2)

so A'H = A'G - HG = 4 - 3 = 1

So the little triangles like A'JK are 1/9 of ABC.

area AJK = 1/9 x 1/9 x 81 = 1

So the required are is 36 - 1 - 1 - 1 = 33

So k = 33/81 = 11/27

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Enshrouded_

Member

Registered: 2015-07-31

Posts: 47

Re: Circle Geo Problems

Can someone help for problem 1?

EDIT: Nevermind, got it

Last edited by Enshrouded_ (2015-09-04 13:15:27)

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: Circle Geo Problems

hi Enshrouded_

To be within 1 unit of the heptagon but not including the heptagon, the locus is the 7 squares shown here in black plus the 7 sectors of a circle.

You can work out the angle of each sector like this:

Where one sector meets two squares and one internal angle of the heptagon, the angles add up to 360.  The internal angle is (2n-4) right angles divided by 7 so do this

There's no need to calculate this angle as 7 of these adds up to a whole circle, radius 1 unit.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob