To solve for the side lengths of each triangle, we start by setting the perimeters for both triangles \(ABC\) and \(PQR\) equal to each other, since they are given to be equal.
Step 1: Find the Perimeter of Triangle ABC
The sides of triangle \(ABC\) are:
- \(AB = 4x + 4\)
- \(BC = 4x\)
- \(AC = 4x + 3\)
The perimeter of triangle \(ABC\) is given by: \[ \text{Perimeter of } ABC = AB + BC + AC = (4x + 4) + (4x) + (4x + 3) \] Combine the terms: \[ \text{Perimeter of } ABC = 12x + 7 \]
Step 2: Find the Perimeter of Triangle PQR
The sides of triangle \(PQR\) are:
- \(PQ = 4x\)
- \(QR = 5x + 5\)
- \(PR = 4x\)
The perimeter of triangle \(PQR\) is given by: \[ \text{Perimeter of } PQR = PQ + QR + PR = (4x) + (5x + 5) + (4x) \] Combine the terms: \[ \text{Perimeter of } PQR = 13x + 5 \]
Step 3: Set the Perimeters Equal
Since the perimeters of the triangles are equal, we have: \[ 12x + 7 = 13x + 5 \]
Step 4: Solve for \(x\)
Subtract \(12x\) from both sides: \[ 7 = x + 5 \]
Subtract \(5\) from both sides: \[ x = 2 \]
Step 5: Find the Side Lengths
Now, substitute \(x = 2\) back into the expressions for the side lengths.
For triangle \(ABC\):
- \(AB = 4(2) + 4 = 8 + 4 = 12\)
- \(BC = 4(2) = 8\)
- \(AC = 4(2) + 3 = 8 + 3 = 11\)
For triangle \(PQR\):
- \(PQ = 4(2) = 8\)
- \(QR = 5(2) + 5 = 10 + 5 = 15\)
- \(PR = 4(2) = 8\)
Final Result
Thus, the side lengths are:
-
For triangle \(ABC\):
- \(AB = 12\)
- \(BC = 8\)
- \(AC = 11\)
-
For triangle \(PQR\):
- \(PQ = 8\)
- \(QR = 15\)
- \(PR = 8\)
Here are the requested results:
- \(AB = 12\)
- \(PQ = 8\)
- \(BC = 8\)
- \(QR = 15\)
- \(AC = 11\)
- \(PR = 8\)