# Calculate the lenght of the height corresponding to the side BC of the ABC triangle , knowing that AB = 13 , AC = 14 , BC = 15

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Since we know all the triangle sides, then we will use Herom formula to calculate the area:

area(a) = sqrt(p(p-AB)(p-BC)(p-AC)

p= perimeter/2 = (15+13+14)/2 = 21

==> a= sqrt(21(21-15)(21-14)(21-13) = sqrt(7056)=84

Now we know tha area, then we will use the following formula:

area(a) = 1/2 * base * height

84 = 1/2 BC * h

==> 84= 1/2 * 15 *h

==> 84*2/15 =h

==> h = 11.2

The area of the triangle is

sqrt {s(s-a)(s-b)(s-c)} where s = (1/2)(a+b+c) =(1/2)(15+14+13) =21

Area = (1/2)21(21-15)(21-14)(21-13) = (1/2)sqrt(21*6*7*8)

But (1/2) BC*h is also the area of the traingle = (1/2) sqrt (21*6*7*8).

Or

h ={ sqrt(21*6*7*8)}/BC = sqrt(21*6*7*8}/15 = 5.6 is the height fro the side BC.

When calculating the area of a triangle, we use the length of the height.

S = (BC*h)/2

The area of a triangle could be calculated also using Heron formula, which in this case, knowing just the sides of the triangle, is more suited:

S = sqrt [p(p-a)(p-b)(p-c)],where p is half-perimeter of the ABC triangle.

p = (a+b+c)/2=(AB+AC+BC)/2=(13+14+15)/2=21

S = sqrt [p(p-a)(p-b)(p-c)]

S = sqrt[p( p-AB)(p-AC)(p-BC)]

S=sqrt[21(21-13)(21-14)(21-15)]=84

84 = (BC*h)/2=(15*h)/2

**h=(2*84)/15=56/5**