the perimeter of a triangle is 56 inches. the longest side measures 4 inches less than the sum of the other two sides. three times the shortest side is 4 inches more than the longest side. find the lengths of the three sides
a+b+c = 54
c = a+b-4
3a = c+4
now start substituting to solve
Let's assume the lengths of the three sides of the triangle are represented by a, b, and c, where a, b, and c are positive numbers.
According to the given information:
1) The perimeter of the triangle is 56 inches, so we have: a + b + c = 56.
2) The longest side (c) measures 4 inches less than the sum of the other two sides (a and b), so we have: c = a + b - 4.
3) Three times the shortest side (a) is 4 inches more than the longest side (c), so we have: 3a = c + 4.
To solve the system of equations, let's substitute the values from equation 2) and 3) into equation 1):
a + b + a + b - 4 = 56,
2a + 2b = 60,
a + b = 30.
Now, we have a new equation: a + b = 30.
Substituting the value of b = 30 - a into equation 2), we have:
c = a + (30 - a) - 4,
c = 30 - 4,
c = 26.
Finally, substituting the value of c = 26 into equation 3):
3a = 26 + 4,
3a = 30,
a = 10.
Now we can find the value of b by substituting the value of a:
b = 30 - a,
b = 30 - 10,
b = 20.
Therefore, the lengths of the three sides of the triangle are a = 10 inches, b = 20 inches, and c = 26 inches.
To find the lengths of the three sides of the triangle, let's assign variables to represent the lengths of the sides. Let's call the lengths of the sides a, b, and c, where a is the longest side.
Based on the given information, we can create the following equations:
1) Perimeter equation: a + b + c = 56 (since the perimeter is given as 56 inches)
2) Longest side equation: a = b + c - 4 (since the longest side measures 4 inches less than the sum of the other two sides)
3) Shortest side equation: 3b = a + 4 (since three times the shortest side is 4 inches more than the longest side)
We have a system of three equations with three variables, so we can solve these equations simultaneously to find the values of a, b, and c.
From equation 2, we can express a in terms of b and c: a = b + c - 4.
Substituting this value of a in equation 3, we get: 3b = (b + c - 4) + 4.
Simplifying equation 3, we get: 3b = b + c.
Subtracting b from both sides, we have: 2b = c.
Now we have a relationship between b and c, which we can use to rewrite equation 1.
Substituting 2b for c in equation 1, we get: a + b + 2b = 56.
Combining like terms, we have: a + 3b = 56.
Now we can use the new equation a + 3b = 56 with equation 2 (a = b + c - 4) to solve for a and b.
Substituting b + c - 4 for a in a + 3b = 56, we get: b + c - 4 + 3b = 56.
Combining like terms, we have: 4b + c - 4 = 56.
Adding 4 to both sides, we get: 4b + c = 60.
Now we have a system of two equations with two variables:
2b = c,
4b + c = 60.
Solving these equations will give us the values of b and c, which we can then substitute back into the equations to find the lengths of the three sides of the triangle.