Introduction:
A triangle is defined as a geometrical figure formed by three non concurrent line segments, which intersect each other. The altitude of a triangle is the perpendicular distance from any of its vertices to the opposite side. The opposite side of the vertex is called as base of the triangle. In simple terms altitude is the height of the triangle. A few properties of the altitude of a triangle are:
We know that the formula to find the area of a triangle is,
A = (1/2)×b×h
Where,
b = base of the triangle
h = altitude of the triangle
Then we can find the altitude as follows:
Altitude (h) = 2A / b
When the length of the three sides of a triangle is called, we can find the altitude of the triangle using heron's formula.
Heron's Formula:
` A = sqrt(s(s-a)(s-b)(s-c))`
`s = (a+b+c )/2`
Where,
A = Area
a, b, c = lengths of the sides
Then we can find the altitude as follows:
Altitude => h = 2A / b
Now clearly the height depends on which side we choose to be the base.
When the triangle is equilateral, then a = b = c Then the formula to find altitude is,
h = (√3 / 2 ) * a
Where,
H = altitude of the equilateral triangle
a = length of each side of the equilateral triangle
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Example 1: The lengths of the 3 sides of a triangle are 8, 6, and 4. Find the altitude
Solution: a = 8, b = 6, c = 4
s = (a + b + c) / 2
s = (8+6+4) / 2 = 9
`A = sqrt(s(s-a)(s-b)(s-c))`
`A =sqrt(9(9-8) (9-6) (9-4))`
`A=sqrt(9*1*3*5)`
`A=sqrt135`
A =11.6 Sq units.
To find the altitude we can use the formula h = 2A / b
The altitude is depending upon on which side we choose to be our
base.
h_{1} = 2(11.6)/8 = 2.9 units
h_{2} = 2(11.6)/6 = 3.86 units
h_{3} = 2(11.6)/4 = 5.8 units
Example 2: Find the altitude of an equilateral triangle having the side of 22 inches.
Solution : The formula to find the altitude is √3/2 * a
Where a = 22 inches
h = ( √3/ 2)* 22
h = 19.05 inches
Hence the altitude is 19.05 inches
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