# Determine the length of the height of triangle ABC, that is perpendicular to BC, if BC= 15, AB=13 and AC=14.

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### 3 Answers

We have the triangle ABC where :

AB = 13 AC = 14 and BC = 15

Then we can find the area usind the formula:

a =sqrt P(p-a)(p-b)(p-c)

Forst let us determine P

P = perimeter/2 = 13+14+15/2 = 42/2 = 21

==> a = sqrt(21*6*7*8) = 84

We also know that:

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

Since we need to determine the height of the perpindicular on BC, then let the height be AD and the base is BC.

==> a = (1/2)*AD*BC

==> 84= (1/2)*AD*15

==> AD = 84*2/15 = 56/5=11.2

Then AD = 11.2

The sides of the triangle a= BC = 15, c = AB = 13 and b = AC = 13.

Therefore area of the triangle sqrt{s(s-a)(s-b) (s-c)}, where s = (a+b+c)/2 = (15+13+14)/2 = 42/2 =21.

Area of the triangle = {sqrt{21(21-15)(21-13)(21_14)} = sqrt{21*6*8*7} = 84 sq units.

If the height h from from BC to A , the rea = (1/2) BC*h = Area = 84

Therefore h = 2*84/BC = 168/15 = 11.2.

We'll apply the area formula:

S = (BC*h)/2, where h is the height perpendicular to BC.

We'll determine the value of the area, using Heron's formula:

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-AB)(p-AC)(p-BC))

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

84 = (BC*h)/2

84 = (15*h)/2

h=(2*84)/15

**h=56/5**