To solve for the lengths of the sides of the isosceles triangle, let's denote the length of the base by [tex]\( b \)[/tex] meters. Consequently, the length of each leg of the triangle is [tex]\( b + 4 \)[/tex] meters since the legs are 4 meters longer than the base.
The perimeter of a triangle is the sum of the lengths of all its sides. For this isosceles triangle, the perimeter comprises the base and the two equal legs. Hence, the perimeter [tex]\( P \)[/tex] can be expressed as:
[tex]\[ P = b + 2(b + 4) \][/tex]
We know that the perimeter of the triangle is 44 meters. Substituting 44 for [tex]\( P \)[/tex], we get:
[tex]\[ 44 = b + 2(b + 4) \][/tex]
Now, let's solve this equation step by step to find the value of [tex]\( b \)[/tex]:
Step 1: Expand the equation:
[tex]\[ 44 = b + 2b + 8 \][/tex]
Step 2: Combine the [tex]\( b \)[/tex] terms on the right-hand side:
[tex]\[ 44 = 3b + 8 \][/tex]
Step 3: Isolate the term containing [tex]\( b \)[/tex] by subtracting 8 from both sides of the equation:
[tex]\[ 44 - 8 = 3b \][/tex]
[tex]\[ 36 = 3b \][/tex]
Step 4: Solve for [tex]\( b \)[/tex] by dividing both sides by 3:
[tex]\[ b = \frac{36}{3} \][/tex]
[tex]\[ b = 12 \][/tex]
The length of the base [tex]\( b \)[/tex] is therefore 12 meters.
Next, we find the length of each leg. Since each leg is [tex]\( b + 4 \)[/tex]:
[tex]\[ \text{Leg} = 12 + 4 \][/tex]
[tex]\[ \text{Leg} = 16 \][/tex]
Now, we verify the result by calculating the perimeter of the triangle using these side lengths:
[tex]\[ \text{Perimeter} = \text{Base} + 2 \times \text{Leg} \][/tex]
[tex]\[ \text{Perimeter} = 12 + 2 \times 16 \][/tex]
[tex]\[ \text{Perimeter} = 12 + 32 \][/tex]
[tex]\[ \text{Perimeter} = 44 \][/tex]
Thus, the perimeter matches the given value, confirming that our solution is correct.
Therefore, the lengths of the sides of the triangle are:
- Base: 12 meters
- Each leg: 16 meters
