Coordinates of a PointDate: 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/ |
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