Answer:
1) S1 = 4·x² - 12·x + 9
S2 = 12·x² - 26·x + 12
S3 = 6·x² - 5·x - 6
2) S = 10·x² - 33·x + 27
Step-by-step explanation:
The length x in cm > 1.5
Segment AB = 2·x - 3, segment BC = 4·x - 6, segment AD = 2·x + 3 and segment BE = 4·x + 1
1) a) The area of the right triangle ABC = S1 = 1/2 × Base × Height
The base of the triangle ABC = Segment BC = 4·x - 6
The height of the triangle ABC = Segment AB = 2·x - 3
Therefore, The area of the triangle ABC = S1 = 1/2 × (4·x - 6) × (2·x - 3) = 1/2 × (8·x² - 12·x - 12·x + 18) = 4·x² - 12·x + 9
S1 = 4·x² - 12·x + 9
b) The area of the rectangle CBFG = S2 = Base × Height
The base of the rectangle CBFG = BC = 4·x - 6
The height of the rectangle CBFG = BF = 3·x - 2
The area of the rectangle CBFG = S2 =(4·x - 6) × (3·x - 2) = 12·x² - 8·x - 18·x + 12
The area of the rectangle CBFG = S2 = 12·x² - 26·x + 12
S2 = 12·x² - 26·x + 12
c) The area of the trapezoid ADEB = S3 = 1/2 × (Long side + Short side) × The height of the trapezoid ADEB
The long side of the trapezoid ADEB = BE = 4·x + 1
The short side of the trapezoid ADEB = AD = 2·x + 3
The height of the trapezoid ADEB = AB = 2·x - 3
The area of the trapezoid ADEB = S3 = 1/2 × (4·x + 1 + 2·x + 3) × (2·x - 3) = 6·x² - 9·x + 4·x - 6
∴ The area of the trapezoid ADEB = S3 = 6·x² - 5·x - 6
S3 = 6·x² - 5·x - 6
2) S = S1 + S2 - S3 = 4·x² - 12·x + 9 + 12·x² - 26·x + 12 - (6·x² - 5·x - 6) = 10·x² - 33·x + 27
