Solving Altitudes Of Triangles (Math Help)

Introduction:

A triangle can be defined as a geometrical figure formed by three lines, which intersect each other and which are not all concurrent. The simplest definition of an altitude of a triangle is the perpendicular distance from any of its vertices to the opposite side. The opposite side of the vertex is known as base of the triangle. A triangle has 3 vertices so, it have 3 altitudes. Altitudes can be generally used to find the triangle's area.

Altitude of the triangle:

The basic thing necessary to find the altitude of the triangle is the area of the triangle and the sides of the triangle.

As, Area = 1/2(Base) x (Altitude)

Using this formula we can calculate Altitude as,

Altitude = 2 x Area/Base

If we don't know the area of the triangle we can calculate it using Heron's formula:

A ='sqrt(s(s-a)(s-b)(s-c))'

s = (a+b+c ) / 2

Where, A = Area

a, b, c = Sides length

We know that the another formula for finding the area of a triangle is,

A = (1/2 ) ? b ? h

Where , b = base

h = height (altitude)

Then we can solve the altitude as follow :

Altitude = h = 2A / b

So, the height depends on the side which we choose to be our base.

When the triangle is equilateral: ,

Then a = b = c

Then the formula to solving the altitudes is,

h =( 'sqrt(3)' / 2 ) * a

Where, h = altitude/Height

a = side of the equilateral triangle

Altitude of an Obtuse triangle:

The method of finding the altitude for an obtuse triangle is same as that discussed above.Two of the Altitudes of an obtuse triangle lie outside to the triangle.

Example Problems:

Ex 1 : The 3 sides of a triangle are having the lengths 10, 12, 14. Find the altitudes.

Sol : Given :- a = 10 ,b = 12 , c = 15

Find the value of s by using the Heron's formula,

s = (a + b + c) / 2

s = (10 + 12+ 14) / 2

s = 18

Then the area of the triangle can be found by Heron's formula:

A ='sqrt(s(s-a)(s-b)(s-c))'

A = 58.78 sq. units

We then know that the another formula to find the area of a triangle is :

A = (1/2)(b)(h)

So, the altitude is:

2A / b = h

The altitude depends on which side we choose to be our base. The three altitudes are:

h1 = 2(58.78)/10 = 11.76 units

h2 = 2(58.78)/12 =9.79 units

h3 = 2(58.78)/14= 8.39 units

Ex 2 : Find the altitudes of an equilateral triangle having the side of 14 inches.

Sol : The formula to find the altitude is : ('sqrt(3)' /2 )* a

Where a = 14inches

H = 'sqrt(3)' / 2 * 14

H = 12.12 inches.

Hence the altitude of given triangle is 12.12 inches.

Since the triangle is an equilateral triangle, so all the three altitudes will have same length.

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