The perimeter of a triangle is 52 centimeters. The longest is 2 centimeters less than the sum of the other 2 sides. Twice the shortest side is 13 centimeters less than the longest side. Find the lengths of each side of the triangle. What are the lengths of the sides from shortest to longest?
shortest --- x
middle -----y
longest ----z
x+y+z= 52
" Twice the shortest side is 13 centimeters less than the longest side" ---> 2x < z by 13
2x + 13 = z
"The longest is 2 centimeters less than the sum of the other 2 sides" ---> z < x+y by 2
z = x+y - 2
y = z-x+2 = (2x+13) - x + 2
= x + 15
in the 1st:
x+y+z=52
x+(x+15) + (2x+13) = 52
4x = 52-28
x = 6 , then y = 21 and z = 25
The sides are 6, 21 and 25
is the sum equal to 52 ? 6+21+25 = 52 , YES
twice the shortest = 12
is 12 less than 25 by 13 ? , YES
longest = 25
2 less than the sum of the other two = 6+21-2 = 25 , YES
All is good
To solve this problem, let's assign variables to the lengths of the sides of the triangle. Let's denote the shortest side as "a," the middle side as "b," and the longest side as "c."
According to the given information:
1. The perimeter of the triangle is 52 centimeters. This means that the sum of all three sides is equal to 52 cm: a + b + c = 52.
2. The longest side (c) is 2 centimeters less than the sum of the other two sides (a + b): c = a + b - 2.
3. Twice the shortest side (2a) is 13 centimeters less than the longest side (c): 2a = c - 13.
Now, we have a system of equations with three unknowns (a, b, c). We can solve this system to find the values of a, b, and c.
First, let's substitute the values from equations (2) and (3) into equation (1).
a + b + (a + b - 2) = 52
2a = (a + b - 2) - 13
By simplifying and rearranging equation (1):
2a + 2b = 54 (equation 4)
a + b = 37 (equation 5)
Now, let's solve equations (4) and (5) simultaneously for the values of a and b.
Subtract equation (5) from equation (4):
2a + 2b - (a + b) = 54 - 37
2a - a + 2b - b = 17
a + b = 17 (equation 6)
Now, we have two simultaneous equations: equation (5) and equation (6).
Solve equations (5) and (6) simultaneously to find the values of a and b.
From equation (5):
a + b = 37
From equation (6):
a + b = 17
If we subtract equation (6) from equation (5):
(a + b) - (a + b) = 37 - 17
0 = 20
We get a contradiction, meaning there is no solution that satisfies both equations. Therefore, there is no valid triangle that can be formed with the given conditions.