Question

the orthocenter of a triangle is at the origin, and two of the vertices of the triangle are at (-5,0) and (3,4) Find the third vertex of the triangle

Answer 1

Answer:

  (3, -6)

Step-by-step explanation:

The orthocenter is the point of intersection of the altitudes of the triangle. That is, each altitude goes through the orthocenter. Hence, the line from any given vertex through the orthocenter is perpendicular to the opposite side.

The attached figure shows the given points labeled A and B. Point A lies on the x-axis, as does the orthocenter, so the side opposite will be a vertical line (perpendicular to the x-axis).

The remaining side of the triangle will be perpendicular to the line from B through the origin. The line through B has slope 4/3, so the line containing side AC will have slope -3/4 and will go through point A. Its point-slope equation will be ...

  y -0 = -3/4(x +5)

We already know the x-coordinate of vertex C is the same as that of vertex B, so is x=3. Using this in the equation for side AC, we find the y-coordinate to be ...

  y = -3/4(3 +5) = -6

The coordinates of point C, the third vertex, are (3, -6).

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Additional comments

The slope of a line is given by ...

  m = (y2 -y1)/(x2 -x1)

When (x1, y1) = (0, 0), this reduces to ...

  m = y2/x2 . . . . . . the relation we used for the slope of line OB

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The slope of a perpendicular line is the opposite reciprocal of the slope of a line. The line perpendicular to the one with slope 4/3 will be -1/(4/3) = -3/4.

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The point-slope form of the equation of a line is ...

  y -k = m(x -h) . . . . . line with slope m through point (h, k)