The medium side of a triangle is 9 more than 1/2 the shortest side,and the longest side is 4 times the shortest side. If the perimeter is 31 in. Find the lengths of the sides of the triangle
let the shortest side be x
middle side = x/2 + 9
longest side = 4x
x + x/2+9 + 4x = 31
solve for x, I suggest multiplying each term by 2 to clear the fraction
To find the lengths of the sides of the triangle, let's denote the shortest side as x.
According to the given information:
- The medium side is 9 more than half the shortest side, which means it is (1/2)x + 9.
- The longest side is 4 times the shortest side, which means it is 4x.
To solve this problem, we can use the perimeter formula for a triangle:
Perimeter = Sum of all sides
In this case, the perimeter is given as 31 inches, so we can set up the equation:
x + ((1/2)x + 9) + 4x = 31
Let's solve this equation step by step:
First, let's simplify the equation:
x + (1/2)x + 9 + 4x = 31
Now, combine like terms:
(1 + 1/2)x + 13x = 31
Next, simplify the coefficients:
(3/2)x + 13x = 31
Now, let's combine the x terms:
(3/2 + 13)x = 31
Simplify the coefficients again:
(29/2)x = 31
To solve for x, we can multiply both sides of the equation by the reciprocal of (29/2), which is 2/29:
((29/2)x) * (2/29) = 31 * (2/29)
Simplify the left side:
x = (31 * 2) / 29
Now, calculate:
x = 62/29
Since the question asks for the length of all sides of the triangle, we can substitute this value back into our original expressions for the other sides.
The medium side is (1/2)x + 9, so substituting x = 62/29, we have:
Medium side = (1/2)(62/29) + 9
Medium side = 31/29 + 9
Medium side = (31 + 261)/29
Medium side = 292/29
The longest side is 4x, so substituting x = 62/29, we have:
Longest side = 4(62/29)
Longest side = 248/29
Therefore, the lengths of the sides of the triangle are:
Shortest side = x = 62/29
Medium side = 292/29
Longest side = 248/29