Orthocenter Coordinates in a Triangle — Practice Geometry Questions

By Allen Ma, Amber Kuang

To find the orthocenter of a triangle, you need to find the point where the three altitudes of the triangle intersect. In the following practice questions, you apply the point-slope and altitude formulas to do so.

Practice questions

Use your knowledge of the orthocenter of a triangle to solve the following problems.

  1. The coordinates of

    image0.png

    are A (0, 2), B (–2, 6), and C (4, 0). Find the coordinates of the orthocenter of this triangle.

  2. The coordinates of

    image1.png

    are A (0, 0), N (6, 0), and D (–2, 8). Find the coordinates ofthe orthocenter of this triangle.

Answers and explanations

  1. (–8, –6)

    The orthocenter of a triangle is the point where the three altitudes of the triangle intersect. An altitude of a triangle is perpendicular to the opposite side. Because perpendicular lines have negative reciprocal slopes, you need to know the slope of the opposite side. Here’s the slope of

    image2.png

    This means that the slope of the altitude to

    image3.png

    needs to be 1.

    The point-slope formula of a line is y y1 = m (x x1), where m is the slope and (x1, y1) are the coordinates of a point on the line. To find the altitude formed when you connect Point A to

    image4.png

    plug in m = –1 and the coordinates of Point A, (0, 2):

    image5.png

    The slope of

    image6.png

    is

    image7.png

    This means that the slope of the altitude to

    image8.png

    The altitude formed when you connect Point C, (4, 0), to

    image9.png

    is

    image10.png

    To find the orthocenter, you need to find where these two altitudes intersect. Set them equal and solve for x:

    image11.png

    Now plug the x value into one of the altitude formulas and solve for y:

    image12.png

    Therefore, the altitudes cross at (–8, –6).

  2. (–2, –2)

    The orthocenter of a triangle is the point where the three altitudes of the triangle intersect. An altitude of a triangle is perpendicular to the opposite side. Because perpendicular lines have negative reciprocal slopes, you need to know the slope of the opposite side. Here’s the slope of

    image13.png

    This means that the slope of the altitude to

    image14.png

    needs to be 1.

    The point-slope formula of a line is y y1 = m (x x1), where m is the slope and (x1, y1) are the coordinates of a point on the line. To find the altitude formed when you connect Point A to

    image15.png

    plug in m = 1 and the coordinates of A, (0, 0):

    image16.png

    Now find the equation for the altitude to

    image17.png

    The slope of

    image18.png

    This means that the slope of the altitude to

    image19.png

    The altitude formed when you connect Point N, (6, 0), to

    image20.png

    To find the orthocenter, you need to find where the two altitudes intersect. Set them equal and solve for x:

    image21.png

    Now plug the x value into one of the altitude formulas and solve for y:

    y = x

    y = –2

    Therefore, the altitudes cross at (–2, –2).