the perimeter of a triangle is 42 inches. three times the length of the longest side minus the length of the shortest side is 49 inches. the sum of the length of the longest side and twice the sum of both the other side lengths is 65 inches. find the side lengths
a = the length of the shortest side
b = the length of the middle side
c = the length of the longest side
P = the perimeter
The perimeter of a triangle is 42 inches.
P = a + b + c
a + b + c = 42 Subtract c to both sides
a + b + c - c = 42 - c
a + b = 42 - c
The sum of the length of the longest side and twice the sum of both the other side lengths is 65 inches.
c + 2 ( a + b ) = 65
c + 2 ( 42 - c ) = 65
c + 2 * 42 - 2 * c = 65
c + 84 - 2 c = 65
- c + 84 = 65 Add c to both sides
- c + 84 + c = 65 + c
84 = 65 + c Subtract 65 to both sides
84 - 65 = 65 + c - c
19 = c
c = 19 in
Three times the length of the longest side minus the length of the shortest side is 49 inches.
3 c - a = 49 Add a to both sides
3 c - a + a = 49 + a
3 c = 49 + a Subtract 49 to both sides
3 c - 49 = 49 + a - 49
3 c - 49 = a
a = 3 c - 49
a = 3 * 19 - 49
a = 57 - 49 = 8
a = 8 in
a + b + c = 42
8 + b + 19 = 42
27 + b = 42 Subtract 27 to both sides
27 + b - 27 = 42 - 27
b = 15 in
The side lengths:
a = 8 in
b = 15 in
c = 19 in
84 = 65 + c Subtract 65 to both sides
84 - 65 = 65 + c - 65
19 = c
c = 19 in
Let's assume the side lengths of the triangle are a, b, and c, with a being the shortest side and c being the longest side.
Based on the given information, we can set up the following equations:
1) Perimeter equation: a + b + c = 42
2) Equation involving the longest side and the shortest side: 3c - a = 49
3) Equation involving the longest side and the sum of the other two sides: c + 2(a + b) = 65
To solve these equations, we can use the method of substitution.
Step 1: Solve equation 2 for a in terms of c.
3c - a = 49
a = 3c - 49
Step 2: Substitute the expression found for a into equations 1 and 3.
For equation 1:
(3c - 49) + b + c = 42
4c + b = 91 (equation 4)
For equation 3:
c + 2((3c - 49) + b) = 65
c + 6c - 98 + 2b = 65
7c + 2b = 163 (equation 5)
Step 3: Solve equations 4 and 5 simultaneously.
Multiply equation 4 by 7 and equation 5 by 4 to eliminate b:
28c + 7b = 637 (equation 6)
28c + 8b = 652 (equation 7)
Subtract equation 6 from equation 7:
(28c + 8b) - (28c + 7b) = 652 - 637
b = 15
Step 4: Substitute the value of b back into equation 4 to find c:
4c + 15 = 91
4c = 91 - 15
4c = 76
c = 19
Step 5: Substitute the values of b and c into equation 1 to find a:
a + 15 + 19 = 42
a = 42 - 15 - 19
a = 8
Therefore, the side lengths of the triangle are: a = 8 inches, b = 15 inches, and c = 19 inches.
To find the side lengths of the triangle, let's assign variables to each side:
Let the longest side be x
Let the other two sides be y and z
We are given the following information:
1. The perimeter of the triangle is 42 inches.
The perimeter of a triangle is calculated by adding up all three side lengths. In this case, we have:
x + y + z = 42 -- equation 1
2. Three times the length of the longest side minus the length of the shortest side is 49 inches.
Mathematically, we can write this as:
3x - z = 49 -- equation 2
3. The sum of the longest side and twice the sum of the other two side lengths is 65 inches.
Mathematically, we can express this as:
x + 2(y + z) = 65 -- equation 3
Now, we have a system of three equations (equations 1, 2, and 3) with three variables (x, y, and z). We can solve this system to find the side lengths.
One way to solve this system is by using the substitution method or the elimination method. For simplicity, let's use the substitution method.
From equation 1, we can express x in terms of y and z:
x = 42 - y - z
Substituting this value of x into equations 2 and 3:
3(42 - y - z) - z = 49 -- equation 4
42 - y - z + 2(y + z) = 65 -- equation 5
Expanding equation 4:
126 - 3y - 3z - z = 49
77 = 4y + 4z
4y + 4z = 77 -- equation 6
Simplifying equation 5:
42 - y - z + 2y + 2z = 65
77 = y + z -- equation 7
Now we have two equations: equation 6 and equation 7. We can solve this system to find the values of y and z.
From equation 7, we can express y in terms of z:
y = 77 - z
Substituting this value of y into equation 6:
4(77 - z) + 4z = 77
308 - 4z + 4z = 77
308 = 77
Since this equation cannot be true, it means there is no solution that satisfies all three equations simultaneously. Therefore, there are no valid side lengths that satisfy the given conditions.