One side of a triangle is 6 meters more than twice the shortest side. The perimeter is 75 meters. Find all three sides.
let shortest side be x
then the described side is 2x+6
third side = y
x + 2x + 6 + y = 75
y = 69 - 3x
restrictions: the sum of any two sides must be greater than the third side
x + 2x+6 > 69-3x ---> 6x > 63 , x > 10.5
x + 69-3x > 2x+6 --->-4x > -63 , x < 15.75
69-3x + 2x+6 > x ---> -2x > 75 , x < -37.5, not possible
10.5 < x < 15.75
try x = 12
one side is 12, the second side is 30, the third side is 69-3(12) = 33
is 12+30>33 , yes
is 12+33> 30 , yes
is 33+30> 12 yes
try x = 14
one side is 14, the second side is 34, the third side is 69-3(14) = 27
is 14+34>27 , yes
is 14+27>34 , yes
is 27+34> 14, yes
as you can see there is an infinite number of solutions, as long as you the smallest side is between 10.5 and 15.75
To solve this problem, let's first represent the sides of the triangle.
Let the length of the shortest side be x meters.
According to the problem, one side of the triangle is 6 meters more than twice the shortest side. So we can write the following equation:
x + (2x + 6) + third side = Perimeter
Since the perimeter is given as 75 meters:
x + (2x + 6) + third side = 75
Now, let's simplify the equation:
3x + 6 + third side = 75
Since we don't have any information about the third side, let's represent it as y:
3x + 6 + y = 75
Rearranging the terms in the equation, we get:
3x + y = 75 - 6
3x + y = 69
Now, we need another equation to solve for both x and y. Let's use the fact that the sum of the lengths of any two sides of a triangle is greater than the length of the third side. In other words:
x + (2x + 6) > y
x + 2x + 6 > y
3x + 6 > y
Now we have the system of equations:
3x + y = 69
3x + 6 > y
To solve this system, we can substitute the second equation into the first equation:
3x + (3x + 6) = 69
Simplifying this equation:
6x + 6 = 69
6x = 69 - 6
6x = 63
x = 63/6
x = 10.5
Now that we have the value of x, we can substitute it into one of the equations to find y:
3(10.5) + y = 69
31.5 + y = 69
y = 69 - 31.5
y = 37.5
So the shortest side of the triangle is 10.5 meters, the second side is 2(10.5) + 6 = 27 meters, and the third side is 37.5 meters.