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?
In triangle ABC, the measure of angle B is 27 degrees more than three times the measure of angle A. The measure of angle C is 48 degrees more than the measure of angle A. Find the measure of each angle.
Let's assume the lengths of the sides of the triangle are a, b, and c, with a being the shortest side.
Given that the perimeter of the triangle is 52 centimeters, we can write the equation:
a + b + c = 52 ...(1)
It is also given that the longest side is 2 centimeters less than the sum of the other two sides, so we can write:
c = a + b - 2 ...(2)
Furthermore, it is given that twice the shortest side is 13 centimeters less than the longest side, so we can write:
2a = c - 13 ...(3)
To solve this system of equations, we will substitute equation (2) and equation (3) into equation (1) and solve for a, b, and c.
Substituting equation (2) and equation (3) into equation (1), we have:
a + b + a + b - 2 = 52
2a + 2b = 54
a + b = 27 ...(4)
Substituting equation (3) into equation (4), we have:
a + (c - 13) = 27
a + c = 40 ...(5)
Now we have a system of two equations (4) and (5) with two variables (a and c). We can solve this system of equations to find the values of a and c.
From equation (5), we can express a in terms of c:
a = 40 - c
Substituting this into equation (4), we have:
40 - c + c = 27
40 = 27
This equation is contradictory, which means there is no solution that satisfies all the given conditions. Therefore, it is not possible to find the lengths of the sides of the triangle.
To solve this problem, we need to use the given information to form equations that represent the relationships between the sides of the triangle. Let's denote the lengths of the sides as follows:
Shortest side = x
Middle side = y
Longest side = z
Based on the given information, we can form the following equations:
1) The perimeter of the triangle is 52 centimeters:
x + y + z = 52
2) The longest side is 2 centimeters less than the sum of the other two sides:
z = x + y - 2
3) Twice the shortest side is 13 centimeters less than the longest side:
2x = z - 13
Now, we have a system of three equations. We can solve this system of equations to find the values of x, y, and z.
First, let's express equation 2) in terms of z:
x + y = z + 2
We can then substitute this expression into equation 1) to eliminate z:
x + y + z + 2 = 52
(z + 2) + z = 52
2z + 2 = 52
2z = 50
z = 25
Now that we have the value of z, we can substitute it back into equation 2) to find x and y:
25 = x + y - 2
27 = x + y
Next, we substitute z = 25 into equation 3) to find x:
2x = 25 - 13
2x = 12
x = 6
Finally, we substitute the values of x and z back into the equation x + y = 27 to find y:
6 + y = 27
y = 21
Therefore, the lengths of the sides of the triangle are:
Shortest side = x = 6 centimeters
Middle side = y = 21 centimeters
Longest side = z = 25 centimeters
So, the lengths of the sides from shortest to longest are 6 centimeters, 21 centimeters, and 25 centimeters.