Final answer:
The perimeter of the isosceles triangle is found by solving for x in the congruent side expressions, substituting back to find the lengths of the sides, and then summing these lengths. The perimeter is 82 units.
Step-by-step explanation:
Finding the Perimeter of an Isosceles Triangle
To find the perimeter of an isosceles triangle where two sides are congruent (AB is congruent to BC), first equalize the expressions for the congruent sides. In this case, AB = 5x + 7 and BC = 12x - 14. Since AB = BC, we can set up the equation 5x + 7 = 12x - 14.
Solving for x, we subtract 5x from both sides and add 14 to both sides, resulting in:
7x = 21
x = 3
Now we can find the lengths of AB and BC by substituting x:
AB = 5x + 7 = 5(3) + 7 = 22
BC = 12x - 14 = 12(3) - 14 = 22
The length of the third side, CA, is given as 4x + 26. We substitute x to find:
CA = 4x + 26 = 4(3) + 26 = 38
To find the perimeter of the triangle, we add up the lengths of all three sides:
Perimeter = AB + BC + CA = 22 + 22 + 38 = 82
The perimeter of the isosceles triangle is 82 units.