Math Is Fun Forum

evene

Member

Registered: 2015-10-18

Posts: 272

Official Geometry Topic (For evene)

Hi, some problems that I don't know how to do. For (1), it says to use Power of a Point

(1) Circles with centers at O and P have radii 2 and 4, respectively, and are externally tangent. Points A and B on the circle with center $O$ and points $C$ and $D$ on the circle with center P are such that AD and BC are common external tangents to the circles. What is the area of the concave hexagon AOBCPD

(2) The length of each edge of a cube is 10 cm. and point K is placed at the center of a face of the cube. A line is drawn through the cube, as shown, from point K to point J, a vertex of the cube on the opposite face. What is the length of KJ? Express your answer in simplest radical form.

(3)

has side lengths x-1, x+1 and x+3. For what value of x is

a right triangle?

evene

Member

Registered: 2015-10-18

Posts: 272

Re: Official Geometry Topic (For evene)

Never mind. Don't do number (3) and (2). I already figured it out. It is easy once you use you brain.

(3)

Solve it for x=7

(2) Drawing a line from K to the vertice above J gives a right triangle. Now use multiple Pythagorean Theorems

Last edited by evene (2015-11-20 11:35:04)

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Official Geometry Topic (For evene)

Hi;

---

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.

evene

Member

Registered: 2015-10-18

Posts: 272

Re: Official Geometry Topic (For evene)

Although (1) is something that I need help on

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Official Geometry Topic (For evene)

Hi;

See of you can work your way to this answer...

---

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.

evene

Member

Registered: 2015-10-18

Posts: 272

Re: Official Geometry Topic (For evene)

Finally, I finished it. I just drew an extra line from O to DP so it's perpendicular!

evene

Member

Registered: 2015-10-18

Posts: 272

Re: Official Geometry Topic (For evene)

Equilateral triangle

and a circle with center O are constructed such that BC is a chord of the circle and point A is the circumcenter of BCO in its interior. If the area of circle with center O is 48 pi, then what is the area of triangle

?

Let ABCD be a square, and let M and N be the midpoints of BC and CD}, respectively. Find sin of angle MAN.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Official Geometry Topic (For evene)

Hi;

Let ABCD be a square, and let M and N be the midpoints of BC and CD}, respectively. Find sin of angle MAN.

I am getting 3 / 5.

---

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: 10,131

Re: Official Geometry Topic (For evene)

hi evene,

The first circle is too large to show fully.  As you know its area you can work out its radius, OB.

The triangle ABO is isosceles so you can split it in half to create a right angle and then use Pythagoras to calculate AB.  From that it should be straight forward to work out the area 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

evene

Member

Registered: 2015-10-18

Posts: 272

Re: Official Geometry Topic (For evene)

How do you get the height of

? When you split it in half, how to you get the height?

Bob

Administrator

Registered: 2010-06-20

Posts: 10,131

Re: Official Geometry Topic (For evene)

Whoops!  Sorry.  Not Pythagoras but trigonometry.  That dotted line cuts 60 in half so angle BAO = 150.  So AB = half OB / cos(15)

Hope that helps,

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

evene

Member

Registered: 2015-10-18

Posts: 272

Re: Official Geometry Topic (For evene)

oh lol ok thanks

Bob

Administrator

Registered: 2010-06-20

Posts: 10,131

Re: Official Geometry Topic (For evene)

Sometimes my brain is so sure I've got it sorted I forget to pay attention to what I'm saying.  I could see a calculation was possible there; just got the wrong one.  I ought to have learnt by now but I haven't.  In fact I seem to be getting worse.  Just keep asking until I start making sense.  .

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

evene

Member

Registered: 2015-10-18

Posts: 272

Re: Official Geometry Topic (For evene)

I have a question about another alternative method I just thought up of:

Let the midpoint of BC be called M and let BC=a

and

. So by Pythagorean Theorem, we have

Note: Refer to Bob's diagram above

Last edited by evene (2015-11-21 14:13:35)

Bob

Administrator

Registered: 2010-06-20

Posts: 10,131

Re: Official Geometry Topic (For evene)

Excellent!!! 

Always better when you can make your own method and I like it better as it doesn't require cos(15).  You meant BO^2 of course and you can say that directly from pi r squared.

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

evene

Member

Registered: 2015-10-18

Posts: 272

Re: Official Geometry Topic (For evene)

Yeah that's what I meant, I'm sure that you got the idea

evene

Member

Registered: 2015-10-18

Posts: 272

Re: Official Geometry Topic (For evene)

Ok so now, I just need help simplifying

EDIT: Noticed a mistake

Last edited by evene (2015-11-21 14:14:05)

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Official Geometry Topic (For evene)

Hi;

Expand the LHS out to:

---

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.

evene

Member

Registered: 2015-10-18

Posts: 272

Re: Official Geometry Topic (For evene)

Need help with this proof problem

Let

denote the circular region bounded by

. The lines

and

partition

into four regions

. Let

denote the area of region

. If

, then compute

.

Last edited by evene (2015-11-22 03:38:18)

evene

Member

Registered: 2015-10-18

Posts: 272

Re: Official Geometry Topic (For evene)

I tried my best to simplify

and this is what I have. Can you verify if it's correct?

Thank you for your patience

EDIT: Changed the mistake pointed out by Bob

Last edited by evene (2015-11-25 05:50:12)

Bob

Administrator

Registered: 2010-06-20

Posts: 10,131

Re: Official Geometry Topic (For evene)

I think that should be 3a^2 over 4 not 2.

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

evene

Member

Registered: 2015-10-18

Posts: 272

Re: Official Geometry Topic (For evene)

Wow... another big bonehead moment for me...

Bob

Administrator

Registered: 2010-06-20

Posts: 10,131

Re: Official Geometry Topic (For evene)

No, no, no.  Not bonehead.  You got a special award (gold star) earlier.  It just seemed to me that all those curves made it impossible, so there ought to be a way to get rid of them.  Did you like the diagram ?  Confucius, he say, "A picture is worth ten thousand words." 

That's twice his name has come up in the last half hour.

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: Official Geometry Topic (For evene)

Reminds me of a story:

Disclaimer: This excerpt is in no way putting down doctors, bigfoot or people in the old country.

  I had just been in a bike accident where I was thrown off my bike at a speed of 30 mph ( I was going downhill ). My head contacted the pavement several times as I skidded down the street. The bike was skidding after me and kabonked me in the head too! I had some bruises and Doctor Vinny Boombotz, was amazed I was still alive. I calmy explained that I had been hit on the head with a pipe before with no effect.

  He scoffed and dragged me over to his x-ray machine where they bombarded my cranium with the deadly rays. The film showed not a single fracture and he said I had an unusually thick skull , similar to a sasquatch skull. Now being compared to a bigfoot is an insult anywhere in the world but I overlooked it. I explained that all the m's were called boneheads back in the old country but we all thought that was just an insult.

Moral: It ain't so bad to be a bonehead.

---

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.

Agnishom

Real Member

From: Riemann Sphere

Registered: 2011-01-29

Posts: 24,974

Website

Re: Official Geometry Topic (For evene)

I have no idea what that means but I will curate it.

---

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.