ALTITUDE
(ORTHOCENTER)
1. Altitude of Triangle: Definition: When altitude is used in math, it's generally referring to the altitude
of a polygon. For instance, in the altitude of a triangle, the altitude refers to the perpendicular
distance from the vertex to the opposite side. In the image, note that the altitude is AD. AD is the
altitude from A to BC.
2. Altitudes in math usually refer to height, however, it will always be the height that is perpendicular to
the base. Some problems in geometry will require you to determine the measurement of the altitude
based on knowing other measurements of the polygon.
3. The orthocenter is not always inside the triangle. If the triangle is obtuse, it will be outside. To
make this happen the altitude lines have to be extended so they cross.
4.
The point where the altitudes of a triangle meet is known as the Orthocenter. It lies inside for an
acute and outside for an obtuse triangle. Altitudes are nothing but the perpendicular line ( AD, BE
and CF ) from one side of the triangle ( either AB or BC or CA ) to the opposite vertex. Vertex is a
point where two line segments meet ( A, B and C ). In the below example, o is the Orthocenter.
5. IMPORTANT REMINDER: The altitude and perpendicular bisector both make 90˚’s and you have
to remember the perpendicular bisector always has to be at the midpoint opposite the vertex. The
altitude will sometimes be at this same point, but more often will not go through the midpoint.
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6. Method to calculate the orthocenter of a triangle:
Lets find with the points A(4,3), B(0,5) and C(3,-6).
a. Step 1:
𝑦 −𝑦
Find the slope of the sides AB, BC and CA using the formula 𝑥2−𝑥1
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Consider the points of the sides to be (𝑥1 , 𝑦1 ) 𝑎𝑛𝑑 (𝑥2 , 𝑦2 ) respectively.
Kindly note that the slope is represented by the letter 'm'.
Slope of AB (m) = 5-3/0-4 = -1/2.
Slope of BC (m) = -6-5/3-0 = -11/3.
Slope of CA (m) = 3+6/4-3 = 9.
b. Step 2:
Now, let’s calculate the slope of the altitudes AD, BE and CF which are perpendicular to BC, CA
and AB respectively. The slope of the altitude = -1/slope of the opposite side in triangle. Or
remember that it is the negative reciprocal by either multiplying by -1 or flip the above and make
sure it is the opposite sign.
Slope of AD = -1/slope of BC = 3/11.
Slope of BE = -1/slope of CA = -1/9.
Slope of CF = -1/slope of AB = 2.
c. Step 3:
Once we find the slope of the perpendicular lines, we have to find the equation of the lines AD,
BE and CF.
There are 2 different Formula to find the equation of orthocenter of triangle. One is the point slope
formula which is (𝑦 − 𝑦1 ) = 𝑚(𝑥 − 𝑥1 )or you can use y-intercept formula which uses the slope
and the point using y = mx + b. Often considered the shorter way. We will use the point slope
formula .
Lets find the equation of the line AD with points (4,3) and the slope 3/11.
y - 3 = 3/11(x - 4)
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By solving the above, we get the equation 3x - 11y = -21 which becomes y = 11 𝑥 +
21
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Similarly, we have to find the equation of the lines BE and CF.
Equation for the line BE with points (0,5) and slope -1/9 = y-5 = -1/9(x-0)
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By solving the above, we get the equation x + 9y = 45 which becomes y = − 9 𝑥 + 5
Equation for the line CF with points (3,-6) and slope 2 = y+6 = 2(x-3)
By solving the above, we get the equation 2x - y = 12 which becomes y = 2𝑥 − 12
d. Step 4:
Find the values of x and y by solving any 2 of the above 3 equations.
In this example, the values of x any y are (8.05263, 4.10526) which are the coordinates of the
Orthocenter(o).
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