Results 1 to 11 of 11

Math Help - Pythagorean theorem and an equilateral triangle

  1. #1
    Junior Member
    Joined
    Nov 2011
    From
    Karachi
    Posts
    36

    Pythagorean theorem and an equilateral triangle

    Let ABC be an equilateral triangle and P be a point inside this triangle such that PA=x, PB=y and PC=z.

    If z^2 = x^2 + y^2, find the length of the sides of triangle ABC in terms of x and y.

    How do we go about this? AFAIK, the Pythagorean theorem can be applied to Right Angle Triangles. I am sure there is a Right Angle Triangle hidden somewhere in there...
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Aug 2007
    From
    USA
    Posts
    3,111
    Thanks
    2

    Re: Pythagorean theorem and an equilateral triangle

    1) It is an equilateral triangle. The Pythagorean Theorem will not do you much direct good.

    2) If you construct a perpendicular on each of the three sides, through point P, you should start seeing Right Triangles in your dreams.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Nov 2011
    From
    Karachi
    Posts
    36

    Re: Pythagorean theorem and an equilateral triangle

    Yes I know I have to do something like that. But where in heavens name should point P be?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Joined
    Dec 2007
    From
    Ottawa, Canada
    Posts
    3,175
    Thanks
    74

    Re: Pythagorean theorem and an equilateral triangle

    Quote Originally Posted by cosmonavt View Post
    Let ABC be an equilateral triangle and P be a point inside this triangle
    such that PA=x, PB=y and PC=z.
    If z^2 = x^2 + y^2, find the length of the sides of triangle ABC in terms of x and y.
    Say you take those 3 : PA, PB and PC, and form a triangle with them, what will you get?
    A right triangle, agree? If not, then all I can say is you need basic help from your teacher.

    OK, do some work:
    draw equilateral triangle ABC, insert point P inside so that AP=3, CP=4 and BP = 5; like:
    Code:
                     A
     
     
                   x=3
     
     
                  P
     
     
           y=4            z=5
     
     
     
    C                                  B
    x^2 + y^2 = z^2
    3^2 + 4^2 = 5^2
    9 + 16 = 25
    25 = 25 ; got that?

    Now go calculate the side length of triangle ABC.
    Once you "see" what's going on, do over in terms of x,y,z.
    Last edited by Wilmer; November 20th 2011 at 10:10 AM.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member
    Joined
    Nov 2011
    From
    Karachi
    Posts
    36

    Re: Pythagorean theorem and an equilateral triangle

    Well, before you edited ur post, I actually did some research on it.

    As a general rule, for every equilateral triangle of side a, the area is given by \frac{\sqrt{3}a^2}{4}

    If we make a point P at any random location in the triangle and draw three lines through it each perpendicular to a side, we end up in something like this (ignore the side lengths 1. Just assume them to be a)



    Now the area of each separate triangle formed is

    \frac{1}{2} x a x h1

    \frac{1}{2} x a x h2 and

    \frac{1}{2} x a x h3

    respectively.

    Hence the equation:
    \frac{\sqrt{3}a^2}{4} = \frac{1}{2} x a x h1 + \frac{1}{2} x a x h2 + \frac{1}{2} x a x h3

    Multiplying both sides by \frac{2}{a}:

    \frac{\sqrt{3}}{2} a = h1 + h2 + h3

    in our case...

    \frac{\sqrt{3}}{2} a = x + y + z

    and the second equation is:

    z^2 = x^2 + y^2

    Now we can solve the two simultaneously to obtain a.

    BUT

    there are infinite values which satisfy the equation z^2 = x^2 + y^2

    Putting those values in the equation \frac{\sqrt{3}}{2} a = x + y + z yields infinite values for a also.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor
    Joined
    Dec 2007
    From
    Ottawa, Canada
    Posts
    3,175
    Thanks
    74

    Re: Pythagorean theorem and an equilateral triangle

    Quote Originally Posted by cosmonavt View Post
    > Well, before you edited ur post, I actually did some research on it.

    The edit was to ADD stuff; all I had before the edit is still there...

    > If we make a point P at any random location in the triangle......
    >there are infinite values which satisfy the equation z^2 = x^2 + y^2

    Yes; BUT in your problem's case, P cannot be at random,
    since z^2 = x^2 + y^2; so x,y,z MUST be sides of a right triangle.

    > Putting those values in the equation \frac{\sqrt{3}}{2} a = x + y + z yields infinite values for a also.

    So what? You get a in terms of x,y,z; that's what you have been asked to do...
    To wrap up, the problem is worded:
    "find the length of the sides of triangle ABC in terms of x and y."
    (notice that "z" is not part of final expression)

    a = side length, k = SQRT(3)/2

    a = [x + y + SQRT(x^2 + y^2)] / k
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Junior Member
    Joined
    Nov 2011
    From
    Karachi
    Posts
    36

    Re: Pythagorean theorem and an equilateral triangle

    Yeah I just realized it :P Thanks man! I totally forgot the "in terms of x and y" part.
    Follow Math Help Forum on Facebook and Google+

  8. #8
    MHF Contributor
    Joined
    Dec 2007
    From
    Ottawa, Canada
    Posts
    3,175
    Thanks
    74

    Re: Pythagorean theorem and an equilateral triangle

    You're doing pretty good...what grade are you in?
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Junior Member
    Joined
    Nov 2011
    From
    Karachi
    Posts
    36

    Re: Pythagorean theorem and an equilateral triangle

    Ahahaha, I am an A level student. U can say grade 12 in my country. But we are not taught much geometry in here. 90% of our syllabus is calculus. So yeah, it is something to be pleased at.
    Follow Math Help Forum on Facebook and Google+

  10. #10
    MHF Contributor
    Joined
    Dec 2007
    From
    Ottawa, Canada
    Posts
    3,175
    Thanks
    74

    Re: Pythagorean theorem and an equilateral triangle

    Quote Originally Posted by cosmonavt View Post
    Ahahaha, I am an A level student. U can say grade 12 in my country. But we are not taught much geometry in here. 90% of our syllabus is calculus. So yeah, it is something to be pleased at.
    "U" is not a word; should be "You":
    may be important if you apply for a job in English !!
    Follow Math Help Forum on Facebook and Google+

  11. #11
    MHF Contributor
    Joined
    Aug 2007
    From
    USA
    Posts
    3,111
    Thanks
    2

    Re: Pythagorean theorem and an equilateral triangle

    Quote Originally Posted by cosmonavt View Post
    If we make a point P at any random location in the triangle
    Another important word to get right.

    You mean "an arbitrary point", not "a point...at any random location".

    "Random" and "arbitrary" often are confused, but they are necessarily different.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. ABC Equilateral Triangle
    Posted in the Algebra Forum
    Replies: 1
    Last Post: May 31st 2010, 07:22 PM
  2. Equilateral triangle
    Posted in the Geometry Forum
    Replies: 2
    Last Post: March 10th 2010, 12:23 PM
  3. Equilateral Triangle
    Posted in the Geometry Forum
    Replies: 4
    Last Post: August 1st 2009, 02:13 AM
  4. Equilateral Triangle
    Posted in the Geometry Forum
    Replies: 4
    Last Post: October 25th 2008, 10:12 PM
  5. Equilateral Triangle
    Posted in the Geometry Forum
    Replies: 1
    Last Post: October 1st 2006, 02:07 PM

Search Tags


/mathhelpforum @mathhelpforum