the perimeter of a triangle is 54 centimeters. the longest side is 6 centimeters less than the sum of the other two sides. Twice the shortest side is 10 centimeters less than the longest side. Find the lengths of each side of the tirangle
shortest side ---- x
longest ----- 2x + 10
let middle side be y
2x + 10 = x + y -6
y = x+16
so x + 2x+10 + x+16 = 54
4x = 28
x = 7
shortest side = 7
longest side = 24
middle side = 23
check: is the sum = 54 ? yes!
is 24 equal to 6 less than 30 ?, yes
is twice 7 10 less than 24 ? yes!
All is good!
Let's assume the lengths of the sides of the triangle are a, b, and c.
We are given the following information:
1) The perimeter of the triangle is 54 centimeters:
Perimeter = a + b + c = 54
2) The longest side is 6 centimeters less than the sum of the other two sides:
c = a + b - 6
3) Twice the shortest side is 10 centimeters less than the longest side:
2a = c - 10
Now, let's solve these equations step by step to find the lengths of each side.
Step 1: Substitute the value of c from equation 2 into equation 3:
2a = (a + b - 6) - 10
Simplify:
2a = a + b - 16
Step 2: Rearrange the equation:
a - b = -16
Step 3: Substitute the value of c from equation 2 into equation 1:
a + b + (a + b - 6) = 54
Simplify:
2a + 2b = 60
Step 4: Divide equation 2 by 2:
a - b = -8 (equation 4)
Step 5: Add equations 4 and 2a + 2b = 60:
(a - b) + (2a + 2b) = -8 + 60
Simplify:
3a = 52
Step 6: Divide both sides of the equation by 3:
a = 17.33 (rounded to two decimal places)
Step 7: Substitute the value of a into equation 4:
17.33 - b = -8
Step 8: Add 8 to both sides of the equation:
17.33 - b + 8 = 0
Simplify:
25.33 - b = 0
Step 9: Subtract 25.33 from both sides of the equation:
-b = -25.33
Step 10: Multiply both sides of the equation by -1:
b = 25.33 (rounded to two decimal places)
Step 11: Substitute the values of a and b into equation 2:
c = 17.33 + 25.33 - 6
Simplify:
c = 36.66 (rounded to two decimal places)
Therefore, the lengths of each side of the triangle are approximately:
a = 17.33 cm
b = 25.33 cm
c = 36.66 cm
To find the lengths of each side of the triangle, let's denote the three sides as x, y, and z, where z is the longest side.
From the given information, we have three equations based on the perimeter and the relationships between the sides:
1. Perimeter: x + y + z = 54 cm
2. Longest side: z = x + y - 6 cm
3. Relationship between the shortest and longest sides: 2x = z - 10 cm
Now we can solve for the lengths of the sides.
Substituting equation 2 into equation 1, we get:
x + y + (x + y - 6) = 54
Simplifying, we have:
2x + 2y - 6 = 54
2x + 2y = 60
x + y = 30 (dividing both sides by 2)
Now we can substitute this value in equation 3:
2x = (x + y) - 10
Substituting x + y = 30, we get:
2x = 30 - 10
2x = 20
x = 10 cm
Substituting the value of x back into equation 2:
z = x + y - 6
z = 10 + y - 6
z = y + 4
Now we can set z = y + 4 as the longest side and x = 10 as the shortest side. Substituting these values in equation 1 to solve for y:
10 + y + (y + 4) = 54
2y + 14 = 54
2y = 54 - 14
2y = 40
y = 20 cm
Finally, substituting the values of x and y back into equation 2 to find z:
z = x + y - 6
z = 10 + 20 - 6
z = 24 cm
So, the lengths of each side of the triangle are x = 10 cm, y = 20 cm, and z = 24 cm.