a musical triangle has a perimeter of 180 mm. the side of middle length is 2 times as long as the shortest side. the longest side is 3 times as long as the middle side. how long is each side
shortest side -- x
middle side ----2x
longest side --- 3(2x) = 6x
x + 2x + 6x = 180
9x=180
x=20
shortest = 20
middle = 40
longest = 120
BTW, no such triangle is possible, even though a mathematical solution exists.
To find the lengths of each side of the musical triangle, let's assign variables for the lengths of the sides. Let's call the shortest side "x" mm.
According to the given information:
- The side of middle length is 2 times as long as the shortest side, so its length is 2x mm.
- The longest side is 3 times as long as the middle side, so its length is 3(2x) = 6x mm.
The perimeter of a triangle is the sum of the lengths of all its sides. In this case, the perimeter is 180 mm. So, we can set up an equation:
x + 2x + 6x = 180
Combining like terms:
9x = 180
Dividing both sides by 9, we get:
x = 180 / 9
x = 20
Now we know that the shortest side is 20 mm. To find the lengths of the other sides:
Middle side = 2x = 2 * 20 = 40 mm
Longest side = 6x = 6 * 20 = 120 mm
Therefore, the lengths of the sides of the musical triangle are:
Shortest side = 20 mm
Middle side = 40 mm
Longest side = 120 mm