Results 1 to 2 of 2

Thread: inequalities with geometry

  1. #1
    Junior Member
    Oct 2007

    Exclamation inequalities with geometry

    So I got this question I couldn't solve. Can anyone help me solve it? Thank you!

    Rectangle ABCD is 10 cm long and 5 cm wide. Point P is located x centimetres from A on side AB. Point Q is located y centimetres from A on side AD. For what values of y is it possible to locate a point R on side BC such that angle QPR=90^o

    Follow Math Help Forum on Facebook and Google+

  2. #2
    Junior Member
    Nov 2008
    Let's imagine the rectangle is in a coordinate system with origin at P. Also, let BR=z. Then, PQ can be expressed as a vector, PQ=(-x,y). Likewise, PR=(10-x,z). Now since QPR is a right angle the dot product of PQ and PR must be zero, so this gives us the following equation:

    (-x,y) \bullet (10-x,z)=0


    z=\frac{x(10-x)}{y} (1)

    Notice from this that the maximum value that x(10-x) can take is 25 (indeed, complete the square to find that x(10-x)=-(x-5)^2+25). Now, the largest possible value for y is 5 (the point Q can't surpass the side of the rectangle). If y=5, the largest possible value for z is then also 5 (since z_{max}=\frac{(x(10-x))_{max}}{5}=\frac{25}{5}=5), meaning that it's possible to find our point R on the side of the rectangle. Thus y=5, regardless of x, is an acceptable value. So now we have our upper limit for y. Now to find the lower limit.

    It's clear that if we decrease y, z increases. We can continue to decrease y all the way until z=5, after that, R surpasses the side of the rectangle, which we don't want. So the lower limit for y must occur when z=5. Going back to the equation I labeled (1), we get that y=\frac{x(10-x)}{5} when z=5.

    So the values for y for which we can find a point R such that QPR=90 are:

    \frac{x(10-x)}{5} \le y \le 5
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Finite Geometry: Young's Geometry
    Posted in the Advanced Math Topics Forum
    Replies: 0
    Last Post: Sep 15th 2010, 08:20 AM
  2. inequalities
    Posted in the Algebra Forum
    Replies: 1
    Last Post: Sep 2nd 2009, 08:33 AM
  3. Modern Geometry: Taxicab Geometry Problems
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: Mar 30th 2009, 08:32 PM
  4. pdf about inequalities
    Posted in the Math Forum
    Replies: 0
    Last Post: Aug 26th 2008, 05:09 AM
  5. Inequalities with geometry
    Posted in the Geometry Forum
    Replies: 3
    Last Post: Dec 6th 2007, 10:56 PM

Search Tags

/mathhelpforum @mathhelpforum