If the coordinates of a square that is inscribed in a circle are (-3,6), (5,6), (-3,-2), and (5,-2) what is the area of the circle?

Question

anonymous · Answer

If a square is inscribed in a circle, then the diagonal of the square is the diameter.

So take any of the two diagonals, say (5,6) and (-3,-2) and find the lngth of this diagonal using the distance formula. So that distance is the diameter.

The radius is half the diameter.

Then use the formula, Area of a circle = pi(radius)^2.

ineedhelp10 · Answer

@Easyaspi314 Then the diameter will be 8?

anonymous · Answer

One sec..let me see.

anonymous · Answer

I get the diagonal equal to sqrt(128) = 8sqrt(2)

anonymous · Answer

so radius is half of 8sqrt(2) = 4 sqrt(2).

anonymous · Answer

Use the distance formula to be convinced that the diameter is sqrt(128).

ineedhelp10 · Answer

so you mean $$sqrt{128}$$

anonymous · Answer

yes, and sqrt 128 = sqrt 64 times sqrt 2 = 8 sqrt 2

ineedhelp10 · Answer

so techinally 128 is the diameter and 64 is the radius?

anonymous · Answer

not 128, sqrt 128 is the diameter!

anonymous · Answer

sqrt 128 is the diameter!

radius is half of sqrt 128 = 4 sqrt(2)

anonymous · Answer

@ineedhelp10

ineedhelp10 · Answer

I dont get how you got 4 sqrt(2)

anonymous · Answer

Do you agree that the diameter is sqrt(128)?

anonymous · Answer

@ineedhelp10

anonymous · Answer

well, I assume you're not here.

ineedhelp10 · Answer

Yeah I agree with that and I am here

ineedhelp10 · Answer

idk how you got 4 as the radius tho?

ineedhelp10 · Answer

I don get why whould you square 64 to get the radius?

ineedhelp10 · Answer

@Easyaspi314

ineedhelp10 · Answer

are you there?

ineedhelp10 · Answer

@e.mccormick can you please explain this? the person who tried explaning it logged off

e.mccormick · Answer

So basically this, right?
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ineedhelp10 · Answer

yeah, i get that part

e.mccormick · Answer

Where the diagonals cross is the center.  That means from the center to the corner is a radius.  Then you can use the radius formula.
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e.mccormick · Answer

radius to area formula, that is.

ineedhelp10 · Answer

Okay.I understand that part too

e.mccormick · Answer

Kk.  So where do you get messed up at?

ineedhelp10 · Answer

So I know the diagonal is 8 right? and after that I used the Pythagorean Theorom to find the diameter which is 128, but im confused on finding the radius

e.mccormick · Answer

The diagonal is $8\sqrt{2}$

ineedhelp10 · Answer

Thats not how they taught it to me in class

e.mccormick · Answer

Take a triangle a, b, c for sides with c as the hypotenuse:

$a^2+b^2=c^2$

But this is frm a square, so a=b.

$a^2+a^2=c^2$

$2a^2=c^2$

$\sqrt{2a^2}=c$

$a\sqrt{2}=c$

Therefore since a is 8, c is $8\sqrt{2}$

e.mccormick · Answer

It is a rule for any 45-45-90 triangle.

ineedhelp10 · Answer

I think you just confused me even more

e.mccormick · Answer

Well, you said you used the Pythagorean.  I did too.  I can do it with numbers and show you also.

ineedhelp10 · Answer

it'll be better to show me with number plz

e.mccormick · Answer

$8^2+8^2=c^2$

$64+64=c^2$

now, I could say that is 128 on the left, but because I know that the square root of 64 is 8, I am going to do it this way instead:

$2\cdot 64=c^2$

Do you get that?

ineedhelp10 · Answer

Why would you go back and multiply 2 by 64?

e.mccormick · Answer

I can write 128 lots of way.  That is one.

$128 = 2\cdot 8\cdot 8$

$128 = 2\cdot 2\cdot 4 \cdot 2\cdot 4 $

$128 = 2\cdot 2\cdot 2\cdot 2 \cdot 2\cdot  2\cdot 2$

$128 = 2^7$

All of them mean the same thing.

ineedhelp10 · Answer

okay, so then square 128 is the diameter?

e.mccormick · Answer

Yes.  But when I simplify that I get $8\sqrt{2}$

Let me work with the all 2s.

$2\cdot 2\cdot 2\cdot 2 \cdot 2\cdot  2\cdot 2=c^2$

$\sqrt{2\cdot 2\cdot 2\cdot 2 \cdot 2\cdot  2\cdot 2}=c$

Now, I can take them out in pairs because I am taking them out from under a square root.

$2\sqrt{2\cdot 2\cdot 2\cdot 2 \cdot 2\cdot \cancel{ 2\cdot 2}}=c$

$2 \cdot 2\sqrt{2\cdot 2\cdot 2\cdot \cancel{ 2\cdot 2}}=c$

$2 \cdot 2\cdot 2\sqrt{2\cdot \cancel{ 2\cdot 2}}=c$

Ah, and $2 \cdot 2\cdot 2=8$ so:

$2 \cdot 2\cdot 2\sqrt{2}=c$ becomes $8\sqrt{2}=c$

And sode c of the triangle is the diametter of the circle.

ineedhelp10 · Answer

This is confusing to me because my geometry teacher didnt teach it this way to us.

e.mccormick · Answer

Hmm...  Well, you have to be able to at least divide it in half.  Or just use the divided form.  $\sqrt{128}$ means the same thing, so if you want it in that form it is OK.  In fact, since you use $\pi r^2$ that is not a bad form.

e.mccormick · Answer

That is all I am sayig.  It is the same number.  Bit of algebra there.

ineedhelp10 · Answer

oh okay thank you. can you explain it this way instead please

e.mccormick · Answer

OK. so if $D=\sqrt{128}$ and $r=\dfrac{D}{2}$, with me there?

ineedhelp10 · Answer

yeah, i get you

e.mccormick · Answer

$r=\dfrac{\sqrt{128}}{2}$

Now it is just plugging that into the area formula.

e.mccormick · Answer

$\pi r^2 = \pi \left(\dfrac{\sqrt{128}}{2}\right)^2$

Can you solve that?

ineedhelp10 · Answer

so the radius will be $$\sqrt{64}$$

ineedhelp10 · Answer

@e.mccormick

e.mccormick · Answer

Ummm...  $\sqrt{64}=8$ and $\dfrac{\sqrt{128}}{2} = \dfrac{11.31}{2}$ which is less than 6.  So they are not the same.

e.mccormick · Answer

For you to divide, it would have to be under a root on the bottom.

ineedhelp10 · Answer

the radius will be 8, correct?

e.mccormick · Answer

No.

e.mccormick · Answer

You can't divide like that.

ineedhelp10 · Answer

When I put $$\sqrt{128}$$ in my calculator I get 11.31

e.mccormick · Answer

Yes.

e.mccormick · Answer

5.655

ineedhelp10 · Answer

I didnt divide by 2 tho and I got the answer

ineedhelp10 · Answer

so bottomline, the radius is 5.65?

e.mccormick · Answer

$r\approx 5.655$ is correct.  But that is not $\sqrt{64}$.  

Now, does your teacher want an exact answer or an approximate?

ineedhelp10 · Answer

I think the exact answer

e.mccormick · Answer

OK.  Then there are two ways you can do that.  Square it first, then divide, or do the division then square it.  Because it is $\pi r^2$ it has to be squared in there somewhere.

ineedhelp10 · Answer

My teacher didnt divide $$\sqrt{128}$$ by 2 in another problem tho. So will you need to divide the sqaure root for this type of problems each time?

e.mccormick · Answer

Yoy can write it as $r=\dfrac{\sqrt{128}}{2}$ and be just fine.  Then when you do $\pi r^2$ the root goes away, but the bottom gets squared too!

e.mccormick · Answer

Also, in the other problem, you may have had the radius.  This time you have the diameter.  So you have to change the diameter to the radius, which is where the over 2 comes from.

ineedhelp10 · Answer

My teacher said the answer was 32pi for the area, is that correct?

e.mccormick · Answer

For this?  Yes.

ineedhelp10 · Answer

I dont get that when plugging it in my calculator though

e.mccormick · Answer

$r=\dfrac{\sqrt{128}}{2}$ means:

$A =\pi r^2 = \pi \left(\dfrac{\sqrt{128}}{2}\right)^2$

Right?

No need for a calculator.

ineedhelp10 · Answer

Oh okay I think I understnad. But how come in other type of problems you leave it in terms or pi?