Answer:
a. triangle: 8x-5; rectangle: 12x+3.
b. 9(8x-5) = 5(12x+3); x = 5
c. 234 cm²
Explanation:
a. The perimeter of a triangle is the sum of the lengths of its sides:
... Pt = (x -3) + (3x -5) + (4x +3) = x(1+3+4) +(-3-5+3)
... Pt = 8x -5
The perimeter of a rectangle is the sum of the lengths of its sides. Since two adjacent sides have the same lengths as their opposite sides, we can add the lengths of adjacent sides and multiply by 2.
... Pr = 2((4x -1/2) + (2x +2)) = 2(x(4+2) + (-1/2+2)) = 2(6x +3/2)
... Pr = 12x +3
b. The ratio of the perimeters is 5/9, so we have ...
... Pt/Pr = 5/9
... 9·Pt = 5·Pr . . . . . . . multiply by 9·Pr
... 9(8x -5) = 5(12x +3) . . . . . . a linear equation relating the perimeters
... 72x -45 = 60x +15 . . . . . . . . eliminate parentheses
... 12x = 60 . . . . . . . . . . . . . . . . add 45-60x
... x = 5 . . . . . . . . . . . . . . . . . . . . divide by 12
c. The area of the rectangle is the product of its length and width.
... A = L·W = ((2x +2) cm) × ((4x -1/2) cm) = ((2·5+2) cm) × ((4·5 -1/2) cm)
... = (12 cm)×(19.5 cm)
... A = 234 cm²