Coordinates of a Point

Date: 6/12/96 at 9:28:16
From: Wong Cheong Siong
Subject: Coordinates of C

Hi Dr. Maths,

ABC is a right-angled triangle labeled counter-clockwise with its 
point C lying on the line y=3x. A is (2,1) and B is (5,5). Find the 
two possible coordinates of C.

Date: 6/12/96 at 13:16:35
From: Doctor Anthony
Subject: Re: Coordinates of C

You have not provided all the information that is required.  There are 
in fact four possible points where C could be, depending on where the 
right-angle is.  If it is at C there are two possibilities, if at A 
there is a third possibility and if at B there is a fourth 
possibility.  I will assume you meant point C, since that gives two 
possibilities.

The easy way to do this is to write down the equation of the circle on 
AB as diameter, and then find the two points where the line y = 3x 
cuts this circle.  These will be the two possible points C.

The distance AB is sqrt((5-2)^2 + (5-1)^2) = sqrt(3^2 + 4^2) = 
sqrt(25) = 5

So radius of the circle is 2.5, and centre of circle is given by 

{(5+2)/2, (5+1)/2)} = {3.5, 3}  

The equation of the circle is therefore

   (x-3.5)^2 + (y-3)^2 = 2.5^2

Now we put y = 3x in this and solve the resulting quadratic to get the 
coordinates of C

               (x-3.5)^2 + (3x-3)^2 = 2.5^2

  x^2 - 7x + 12.25 + 9x^2 - 18x + 9 = 6.25

                   10x^2 - 25x + 15 = 0

                      2x^2 - 5x + 3 = 0 

                        (2x-3)(x-1) = 0 

The two possible answers are x = 3/2 and x = 1
Corresponding y values are   y = 9/2 and y = 3

Thus C has two possible values:

       (1, 3)  or  (3/2, 9/2)

-Doctor Anthony,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 6/12/96 at 23:42:51
From: Wong Cheong Siong
Subject: Re: Maths Question.

Thank you Dr. Maths. Your help is greatly appreciated. However, this 
method of solving has not been taught yet, so would you explain to me 
more clearly how to apply this method of solving? I do not understand 
where you will draw the circle and whether it will be a full or 
semi-circle and where the circle will be touching which points. Your
answer to this question is correct but I wish you could explain more 
clearly.

Best regards,
Wong Cheong Siong

Date: 6/13/96 at 10:53:6
From: Doctor Anthony
Subject: Re: Maths Question.

The circle is drawn on AB as diameter, and this circle will cut the 
line  y = 3x in two positions, giving the possible positions of 
point C.  As you know, the angle subtended by the diameter of a circle 
at any point on the circumference is a right-angle, so the condition 
that the angle at C is a right-angle is guaranteed.

If you don't like the circle method for solving this problem, here is 
an alternative method.

Let the point C have coordinates (k, 3k) where k is to be found.  We 
require the line CA and CB to be perpendicular so we write down this 
condition in terms of k.  

Slope of CA = (3k-1)/(k-2)     Slope of CB = (3k-5)/(k-5)

Condition for perpendicular lines is that product of slopes equals -1.

So we require {(3k-1)(3k-5)}/{(k-2)(k-5)} = -1

            9k^2 - 18k + 5 = -{k^2 - 7k + 10}

          10k^2 - 25k + 15 = 0 

             2k^2 - 5k + 3 = 0

               (2k-3)(k-1) = 0

So k has the values 3/2 and 1

This gives two positions for C, namely  (3/2, 9/2) and (1, 3)   

-Doctor Anthony,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/