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Question 1006338: Please help me solve the problem
two vertices of a triangle(5,-1) and (-2,3) and if the orthocentre lies on origin Find the third vertex:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the third vertex is at (-4,-7)
how to find it.
first understand what orthocenter means.
it is the intersection of all the altitudes of the triangle.
the altitudes of the triangle pass through one vertex and are perpendicular to the opposite side.
the rest if finding the equations of the lines of the triangles and of the altitudes of the triagnles and then finding the intersections of a couple of those lines to find the third point.
the specific application strategy is shown below:
first get the equation of the line through the points (-2,3) and (5,-1)
the equation of that line is y = (-4/7) * x + 13/7.
next find the line that is perpendicular to that line and passing through the point (0,0).
the equation of that line is y = (7/4) * x.
next find the equation of the line through the point (5,-1) and the point (0,0).
the equation of that line is y = (-1/5) * x.
next find the line that is perpendicular to y = (-1/5) * x and passing through the point (-2,3).
the equation of that line is y = 5 * x + 13.
next find the intersection of y = 5 * x + 13 with y = (7/4) * x.
that intersection is the point (-4,-7).
that's all you need.
as a bonus, i found the equation of the line through the points (-4,-7) and (5,-1) as well.
that equation is y = (2/3) * x - 13/3
the equation of the line through the origin and through the point (5,-1) is equal to y = (-3/2) * x.
all of this is shown on the following graph:
the equations of the lines that form the sides of the triangle are shown in black.
the equations of the lines that are perpendicular to those lines and passing through the origin are shown in red.
basic concepts used:
slope intercept equation of a straight line is y = mx + b
m is the slope
b is the y-intercept.
slope is derived from 2 points on the line.
first point is (x1,y1)
second point is (x2,y2)
slope = (y2-y1) / (x2-1)
slope of a line perpendicular to another line is the negative reciprocal of the slope of that line.
if the slope of the line is -4/7, then the slope of the line perpendicular to it would be 7/4.
if you have the slope of a line and a point on that line, then replace x with the x-coordinate of the point and replace y with the y-coordinate of the point and then solve for b.
example:
y = mx + b.
slope is 2/3.
y = mx + b becomes y = (2/3) * x + b.
point is (5,-1).
y = (2/3) * x + b becomes -1 = (2/3) * 5 + b.
simplify to get -1 = 10/3 + b.
subtract 10/3 from both sides of the equation to get:
-1 - 10/3 = b.
solve for b to get b = -13/3.
y = (2/3) * x + b becomes y = (2/3) * x - 13/3.
if you are confused by how any of this was derived, let me know and i'll do my best to clear up the confusion.
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