how to find vertices of a triangle using midpoint
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Using Midpoints to Determine Vertices
Date: 09/04/2002 at 21:01:01 From: Leanne Subject: Using midpoints to determine vertices We have been learning how to calculate the length of line segments, and one of our questions read like this: The midpoints of the sides of a triangle have coordinates G(3,1), H (-1,2) and J (1,-3). Determine the coordinates of the vertices of the triangle. Help, Dr. math! Date: 09/05/2002 at 11:31:45 From: Doctor Greenie Subject: Re: Using midpoints to determine vertices Hi, Leanne - Let the vertices of the triangle be A, B, and C, such that given point G(3,1) is the midpoint of AB, given point H(-1,2) is the midpoint of AC, and given point J(1,-3) is the midpoint of BC. Let the points A, B, and C be defined by A(x1, y1) B(x2, y2) C(x3, y3) Now, using the known coordinates of the midpoints G, H, and J, the midpoint formula gives us the following equations relating the x- and y-coordinates of points A, B, and C: (x1+x2)/2 = 3; (y1+y2)/2 = 1 [G(3,1) is the midpoint of AB] (x1+x3)/2 = -1; (y1+y3)/2 = 2 [H(-1,2) is the midpoint of AC] (x2+x3)/2 = 1; (y2+y3)/2 = -3 [J(1,-3) is the midpoint of BC] The three equations in x1, x2, and x3 give us a system of equations that we can solve to find the values of x1, x2, and x3; and likewise the three equations in y1, y2, and y3 give us a system of equations that we can solve to find the values of y1, y2, and y3.
Date: 09/04/2002 at 21:01:01 From: Leanne Subject: Using midpoints to determine vertices We have been learning how to calculate the length of line segments, and one of our questions read like this: The midpoints of the sides of a triangle have coordinates G(3,1), H (-1,2) and J (1,-3). Determine the coordinates of the vertices of the triangle. Help, Dr. math! Date: 09/05/2002 at 11:31:45 From: Doctor Greenie Subject: Re: Using midpoints to determine vertices Hi, Leanne - Let the vertices of the triangle be A, B, and C, such that given point G(3,1) is the midpoint of AB, given point H(-1,2) is the midpoint of AC, and given point J(1,-3) is the midpoint of BC. Let the points A, B, and C be defined by A(x1, y1) B(x2, y2) C(x3, y3) Now, using the known coordinates of the midpoints G, H, and J, the midpoint formula gives us the following equations relating the x- and y-coordinates of points A, B, and C: (x1+x2)/2 = 3; (y1+y2)/2 = 1 [G(3,1) is the midpoint of AB] (x1+x3)/2 = -1; (y1+y3)/2 = 2 [H(-1,2) is the midpoint of AC] (x2+x3)/2 = 1; (y2+y3)/2 = -3 [J(1,-3) is the midpoint of BC] The three equations in x1, x2, and x3 give us a system of equations that we can solve to find the values of x1, x2, and x3; and likewise the three equations in y1, y2, and y3 give us a system of equations that we can solve to find the values of y1, y2, and y3.
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