Here we are going to see how to find orthocenter of a triangle with given vertices.

**What is Orthocenter ?**

It can be shown that the altitudes of a triangle are concurrent and the point of concurrence is called the orthocentre of the triangle.The orthocentre is denoted by O.

Let ABC be the triangle AD,BE and CF are three altitudes from A, B and C to BC, CA and AB respectively.

**Steps Involved in Finding Orthocenter of a Triangle :**

- Find the equations of two line segments forming sides of the triangle.
- Find the slopes of the altitudes for those two sides.
- Use the slopes and the opposite vertices to find the equations of the two altitudes.
- Solve the corresponding x and y values, giving you the coordinates of the orthocenter.

**Example 1 :**

Find the coordinates of the orthocentre of the triangle whose vertices are (3, 1), (0, 4) and (-3, 1)

**Solution :**

Now we need to find the slope of AC. From that we have to find the slope of the perpendicular line through B.

Slope of AC = (y_{2} - y_{1}) / (x_{2} - x_{1})

A (3, 1) and C (-3, 1)

here x_{1 }= 3, y_{1} = 1, x_{2 }= -3 and y_{2} = 1

= (1 - 1) / (-3 - 3)

= 0

Slope of the altitude BE = -1/ slope of AC

= 1/0

Equation of the altitude BE :

(y - y_{1}) = m (x -x_{1})

Here B (0, 4) m = 1/0

(y - 4) = (1/0) (x - 0)

(y - 4) = x / 0

x = 0

Now we need to find the slope of BC. From that we have to find the slope of the perpendicular line through D.

Slope of BC = (y_{2} - y_{1}) / (x_{2} - x_{1})

B (0, 4) and C (-3, 1)

here x_{1 }= 0, y_{1} = 4, x_{2 }= -3 and y_{2} = 1

= [(1 - 4) / (-3 - 0)]

= -3 / (-3)

= 1

Slope of the altitude AD = -1/ slope of AC

= -1/1

= -1

Equation of the altitude AD :

(y - y_{1}) = m (x -x_{1})

Here A(3, 1) m = -1

(y - 1) = -1(x - 3)

y - 1 = -x + 3

x + y = 3 + 1

x + y = 4 --------(1)

Substitute the value of x in the first equation

0 + y = 4

y = 4

So the orthocentre is (0, 4).

**Example 2 :**

Find the co ordinates of the orthocentre of a triangle whose vertices are (2, -3) (8, -2) and (8, 6).

**Solution :**

Let the given points be A (2, -3) B (8, -2) and C (8, 6)

Now we need to find the slope of AC.From that we have to find the slope of the perpendicular line through B.

Slope of AC = (y_{2} - y_{1}) / (x_{2} - x_{1})

A (2, -3) and C (8, 6)

here x_{1 }= 2, y_{1} = -3, x_{2 }= 8 and y_{2} = 6

= (6 - (-3)) / (8 - 2)

= 9/6

= 3/2

Slope of the altitude BE = -1/ slope of AC

= -1 / (3/2)

= -2/3

Equation of the altitude BE :

(y - y_{1}) = m (x - x_{1})

Here B (8, -2) m = 2/3

y - (-2) = (-2/3) (x - 8)

3(y + 2) = -2 (x - 8)

3y + 6 = -2x + 16

2x + 3y - 16 + 6 = 0

2x + 3y - 10 = 0

Now we need to find the slope of BC. From that we have to find the slope of the perpendicular line through D.

Slope of BC = (y_{2} - y_{1}) / (x_{2} - x_{1})

B (8, -2) and C (8, 6)

here x_{1 }= 8, y_{1} = -2, x_{2 }= 8 and y_{2} = 6

= (6 - (-2)) / (8 - 8)

= 8/0 = undefined

Slope of the altitude AD = -1/ slope of AC

= -1/undefined

= 0

Equation of the altitude AD:

(y - y_{1}) = m (x - x_{1})

Here A(2, -3) m = 0

y - (-3) = 0 (x - 2)

y + 3 = 0

y = -3

Substitute the value of x in the first equation

2x + 3(-3) = 10

2x - 9 = 10

2x = 10 + 9

2x = 19

x = 19/2

So, the orthocentre is (19/2,-3).

After having gone through the stuff given above, we hope that the students would have understood how to find the orthocenter of a triangle with given vertices.

If you need any other stuff in math, please use our google custom search here.

HTML Comment Box is loading comments...

You can also visit the following web pages on different stuff in math.

**WORD PROBLEMS**

**Word problems on simple equations **

**Word problems on linear equations **

**Word problems on quadratic equations**

**Area and perimeter word problems**

**Word problems on direct variation and inverse variation **

**Word problems on comparing rates**

**Converting customary units word problems **

**Converting metric units word problems**

**Word problems on simple interest**

**Word problems on compound interest**

**Word problems on types of angles **

**Complementary and supplementary angles word problems**

**Trigonometry word problems**

**Markup and markdown word problems **

**Word problems on mixed fractrions**

**One step equation word problems**

**Linear inequalities word problems**

**Ratio and proportion word problems**

**Word problems on sets and venn diagrams**

**Pythagorean theorem word problems**

**Percent of a number word problems**

**Word problems on constant speed**

**Word problems on average speed **

**Word problems on sum of the angles of a triangle is 180 degree**

**OTHER TOPICS **

**Time, speed and distance shortcuts**

**Ratio and proportion shortcuts**

**Domain and range of rational functions**

**Domain and range of rational functions with holes**

**Graphing rational functions with holes**

**Converting repeating decimals in to fractions**

**Decimal representation of rational numbers**

**Finding square root using long division**

**L.C.M method to solve time and work problems**

**Translating the word problems in to algebraic expressions**

**Remainder when 2 power 256 is divided by 17**

**Remainder when 17 power 23 is divided by 16**

**Sum of all three digit numbers divisible by 6**

**Sum of all three digit numbers divisible by 7**

**Sum of all three digit numbers divisible by 8**

**Sum of all three digit numbers formed using 1, 3, 4**

**Sum of all three four digit numbers formed with non zero digits**