All Answers 356759 Answers

AB.CD + BC.AD = AC.BD

is known as Ptolemy's theorem,

dates back around two thousand years.

"Triangle ABC has vertices A (0,0), B (0,3), C(5,0). A point  inside the triangle is  sqrt 10 units from point  A and sqrt 13 units from point B. How many units is P from point C? Express your answer in simplest radical form."

time = distance/speed

speed in still water = v

net speed upstream = v - 5

net speed downstream = 5 + v

So:  2/(v-5) = 12/(5 + v)

Can you take it from here? 

1 x 10^​51

That is what I thought.

I am sure these pricey courses are fantastic for some students.

But for most students they would not be. 

The work would have to be presented at the ideal level and the student would need to be very diligent.

There would also need to be a good help system.  Even if it is only talking to and learning with/from other students. 

To learn anything new you need to be able to ask your own questions and learn directly from other people.

This is my experience anyway.

Otherwise it would be like an unused gym membership.  A total waste of money.

Mmm 

You are using terms and formula that I have not seen before.

But yes, this is probably another way to skin the cat.  (sorry feline lovers - it is just an expression)

ax^2 + bx + c = y       when x = 0       c = 4   (given as the first point)

ax^2 + bx + 4 = y     sub in    3,19

9a + 3b + 4 = 19              and sub in 4, 68

16a + 4b + 4 = 68          now you have two equations and two unknowns  ....do you know how to solve these?

                                          various ways   ..... multiply first equation by - 1 1/3

-12a -4b - 16/3 =  -76/3       add this to the second equation

4a           -4/3 = 128/3

4a = 132/3

a = 11                     then b = -28

11 x^2     - 28x   +4    = f(x)  

Here is a graph:

https://www.desmos.com/calculator/bdjis3s6tt

From first equation     y/x  = (1+y^2) /250

From second equation   y/x = (1-y^2)/ (-240)          Equate these two equations  to solve for y = +- 7

                                                                              sub in these values of y to calculate possible values of 'x'

Sum of interior angles of a polygon =   ( n-2)(180)

so     (n-2)(180) / n = 144

           180n - 360 = 144n

              36n = 360

                   n = 10      number of sides

a1 = 9  (given)    

           add 9 to get a2 = 18   

                         add 9 to get a3  

                                              etc etc

5000 seats

x = 26 dollar seats

5000-x = 40 dollar seats

26 x    + 40 ( 5000 - x)   = 145400        solve for 'x' the # of 26 dollar seats then # of 40 dollar seats = 5000-x

Divide  7x2 +1   into the area

                               9x

7x² +1  |  63 x³   +  9x

               63 x³    + 9x

                              0                      9x = width

There's more than one way to... 

.................................................................

We can use this formula to calculate the circumradius. (R-radius, s-semiperimeter, a,b,c,d-sides)

When we have the length of a circumradius, we can calculate the angles B and C or A and D, and then using the law of cosines we can find diagonals AC and BD.

This is perhaps the longer way, but it would work. 

there are 12 choose 5 = 792 possible ways of getting to point (5,7) from (0,0), and there are 7 choose 3=35 possible ways of getting to point (4,3) from (0,0), so the number of ways to get to (5,7) from (0,0) are

 792-35=$\boxed{757}$

Good idea; each course lasts around 2-3 months(I just did an intro to geom that lasted half a year), and is pretty pricey, a few hundred dollars ;-; 

Welcome aboard, Lurk-ster !     I am sure your contributions will be appreciated ! 

It's called the Law of Cosines,

Hi Oofpirate.

What a nice introduction.  

It will be good to get to know you .   

Please do not post questions from week 4 of Introduction to number theory. 

The distance formula is essentially the Pythagorean theorem, so exactly what you did  

Use tangent, Opposite/adjacent. 

16/10, which is equal to 1.6. Use the inverse of tan, to get the answer of 57.99 which is equal to 58. 

Sorry for replying late, confused myself by using radian mode instead of decimal. :(

Hey there!

Here you want to use Tangent, or Tan for short, and rearrange to give your angle.

\(Tan(\theta)=\frac{Opposite}{Adjacent}=\frac{16}{10}\)

\(\theta =Tan^{-1}(\frac{16}{10}) \)

\(\theta\approx58 degrees\)

Hope this helped :)

Hey there,

Finding \(VT\) and \(UV\) is quite simple as it will just be half of \(RT\) and \(US\) respectively, so:

\(VT=11units\) and \(UV=9units\), (thus \(RV=11units\) and \(VS=9units\))

Finding \(RS\) and \(ST\) will require Pythagoras (\(C=\sqrt{A^2 + B^2}\)). See the working below:

\(RS=\sqrt{(RV)^2+(VS)^2}=\sqrt{11^2 +9^2}\approx14.213units\)
 

Since \(RV=VT\), \(RS=RT\)

Hope this helped :)

Thanks Dragen   

I really do think that there must be a better way to tackle this problem though.