Question

Q1. The perimeter of triangle is 37cm , if the length of the altitudes are 4x , 5x ,6x . Find the lengths of the sides of the triangle
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Answer 1

Answer:

a = 10 units , b = 12 units and c = 15 units

Step-by-step explanation:

Let Given a triangle ABC and its sides AB = c , BC = a , AC = b

Now we are given that altitudes AD : BE : CF are in ratio of 6 : 5 : 4

So let AD = 6x , BE = 5x and CF = 4x where x is some positive integer

Now area of a triangle is ½ × base × height

and with different bases and their corresponding altitudes

we can write area of Δ ABC in three ways

So area of Δ ABC = ½ × BC × AD = ½ × AC × BE = ½ × AB × CF

or ½ × a × 6x = ½ × b × 5x = ½ × c × 4x

Cancelling ½ × x from all three parts we get

6 a = 5 b = 4 c

now we can let 6 a = 5 b = 4 c = k where k is some positive integer

So a = k / 6 , b = k / 5 and c = k / 4

Since perimeter of Δ ABC = 37 given

so we have a + b + c = 37

or we have k / 6 + k / 5 + k / 4 = 37

k ( 1/6 + 1/5 + 1/ 4 ) = 37

k ( 10+12+15)/60 = 37 Taking LCM and adding

k 37 /60 = 37

or we have k = 60

So a = k/6 = 60/6 = 10 , b = k / 5 = 60 / 5 = 12

and c = k / 4 = 60 / 4 = 15

So final answer is a = 10 units , b = 12 units and c = 15 units

                             I hope this answer will help you..............