The sides of triangle are in ratio 5:3:4 and it's perimeter is 180m find altitude of the triangle corresponding to its largest side
this is just a right triangle with longest leg length 4. (Hypotenuse is 5)
5+3+4 = 12
180/12 = 15
So, scale it up by a factor of 15, and its sides are
75:45:60
To find the altitude of the triangle corresponding to its largest side, we can use the properties of a triangle.
Let's assume the sides of the triangle are 5x, 3x, and 4x. We know that the perimeter of the triangle is 180m. So we can write the equation:
5x + 3x + 4x = 180
Simplifying the equation, we have:
12x = 180
Dividing both sides of the equation by 12, we get:
x = 15
Now, we can find the lengths of the sides of the triangle:
5x = 5 * 15 = 75
3x = 3 * 15 = 45
4x = 4 * 15 = 60
We can see that the largest side of the triangle is 75m. To find the altitude of the triangle corresponding to its largest side, we can use the formula:
Area = (1/2) * base * height
Since the largest side is the base of the triangle, we have:
Area = (1/2) * 75 * height
The area of a triangle can also be calculated using Heron's formula, which does not require the height. Heron's formula is:
Area = √(s * (s - a) * (s - b) * (s - c))
where s is the semiperimeter of the triangle, and a, b, and c are the lengths of the sides. In our case, we can calculate the semiperimeter as:
s = (75 + 45 + 60) / 2 = 90
Now, we can calculate the area using Heron's formula:
Area = √(90 * (90 - 75) * (90 - 45) * (90 - 60)) = √(90 * 15 * 45 * 30) = √(3 * 15 * 3 * 3 * 15 * 30)
Since the area of a triangle can also be calculated as (base * height) / 2, we can equate the two expressions for the area:
(75 * height) / 2 = √(3 * 15 * 3 * 3 * 15 * 30)
Simplifying the equation, we have:
(75 * height) / 2 = √(3 * 3 * 3 * 3 * 2 * 5 * 15 * 15)
(75 * height) / 2 = 9 * 3 * √(2 * 5 * 15 * 15)
(75 * height) / 2 = 9 * 3 * 15 * √2
(75 * height) / 2 = 405 * √2
To isolate the height, we multiply both sides of the equation by 2 and then divide by 75:
height = (405 * √2 * 2) / 75
height ≈ 8.56 m
Therefore, the altitude of the triangle corresponding to its largest side is approximately 8.56 meters.