Final answer:
To calculate the perimeter of triangle ABC, the value of x is determined by solving the equation 3x - 5 = 19 - x, yielding x = 6. Substituting x into the given expressions for the sides of the triangle, the lengths become AB = AC = 13 cm and BC = 12 cm, resulting in a total perimeter of 38 cm.
Step-by-step explanation:
To find the perimeter of triangle ABC, we need to add together the lengths of all three sides. Since angle ABC is equal to angle BCA, the triangle is isosceles and sides AB and AC are of equal length, so we can set (3x - 5) equal to (19 - x) to solve for x. Solving the equation:
3x - 5 = 19 - x
4x = 24
x = 6
Now we can find the lengths of each side:
- AB = 3x - 5 = 3(6) - 5 = 18 - 5 = 13 cm
- AC = 19 - x = 19 - 6 = 13 cm
- BC = 2x = 2(6) = 12 cm
Finally, to find the perimeter, we add the lengths:
Perimeter = AB + AC + BC = 13 cm + 13 cm + 12 cm = 38 cm