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Question Number 189558 by normans last updated on 18/Mar/23

	Answered by a.lgnaoui last updated on 19/Mar/23

	$∙ (B C E F) e s t u n t r a p e z e$ $A i r e (B C E F) = (\frac{B C + E F}{2}) \times C E$ $= \frac{(2, 5 + y) (x + 2)}{2} = A_{1} + A_{2} + 9 (1)$ $A_{1} = \frac{2, 5 x}{2}$ $\Rightarrow \frac{(2, 5 + y) (x + 2)}{2} = \frac{2, 5}{2} x + 9 + A_{2}$ $\Rightarrow [y (x + 2) = 13 + 2 A_{2}]$ $2 ∙ (A B G F)$ $A i r e = A_{4} + A_{5} + 9 (A i r e (H I F) = 9)$ $= \frac{(A B + F G)}{2} \times A G = \frac{(y + 7, 5) (x + 2)}{2}$ $A i r e (A B G F) = \frac{x y + 2 y + 7, 5 x + 15}{2} (2)$ $A_{4} = A i r e [(A H C D) - (A_{1} + 9)]$ $A i r e (A H C D) = \frac{(3 + x) (y + 5)}{2}$ $\Rightarrow A_{4} = A i r e (A H B I) = \frac{(3 + x) (y + 5)}{2} - (9 + \frac{2, 5}{2} x)$ $= \frac{3 y + 15 + x y + 5 x}{2} - \frac{18 + 2, 5 x}{2} = \frac{(2, 5 x + 3 y - 3) + x y}{2}$ $A_{4} = \frac{3 y + x y + 2, 5 x - 3}{2}$ $A_{3} = \frac{5 (x - 1)}{2}$ $A i r e (A B G F) = \frac{2 y + 7, 5 x + 15 + x y}{2} = 9 + \frac{7, 5 x + 3 y + x y - 8}{2}$ $y - 5 = 0 \Rightarrow y = 5$ $A i r e d u r e c t a n g l e = (2 + x) (y + 5)$ $= A i r e (A B G F) + i r e (B C F E) : (1 + 2)$ $= \frac{2 y + x y + 7, 5 x + 15}{2} + \frac{2, 5 x + 5 x + 5 + x y}{2}$ $= \frac{15 x + 20 + 2 x y}{2} = \frac{25 x + 20}{2}$ $A i r e r e c t a n g l e = 12, 5 x + 10 (3)$ $12, 5 x + 10 = 10 x + 20$ $2, 5 x = 10 \Rightarrow x = 4$ $d o n c (x, y) = (4, 5)$ $\Rightarrow A i r e (r e c t s n g l e) =$ $(2 + x) (5 + y) =$ $(4 + 2) (5 + 5) = 60$ $- - - - - -$ $A i r e (R e c t a n g l e) = 60$
	Commented by a.lgnaoui last updated on 19/Mar/23

	Answered by mr W last updated on 19/Mar/23

	Commented by mr W last updated on 19/Mar/23

	$0.5 \times 3 \times (2.5 + x) + 0.5 \times 5 y + A + 9 = 0.5 \times (5 + 2.5 + x) (3 + y)$ $\Rightarrow 2.5 y + x y = 3 + 2 A . . . (i)$ $0.5 \times 2.5 \times (y + 1) + 9 + B + 0.5 \times 2 x = 0.5 \times (2.5 + x) (3 + y)$ $\Rightarrow x + x y = 13 + 2 B . . . (i i)$ $0.5 \times 3 \times (2.5 + x) + A + 9 + 0.5 \times 2.5 \times (y + 1) = 0.5 \times (3 + y + 1) (5 + x)$ $\Rightarrow x + 2.5 y + x y = 8 + 2 A . . . (i i i)$ $0.5 \times 5 y + 9 + B + 0.5 \times 2 x = 0.5 \times (2 + y) \times (5 + x)$ $\Rightarrow x y = 8 + 2 B . . . (i v)$ $(i i i) - (i) o r (i i) - (i v) :$ $\Rightarrow x = 5$ $f r o m (i) :$ $7.5 y = 3 + 2 A \Rightarrow A = \frac{7.5 y - 3}{2}$ $f r o m (i v) :$ $5 y = 8 + 2 B \Rightarrow B = \frac{5 y - 8}{2}$ $\frac{A}{9} = \frac{9}{B} \Rightarrow A \times B = 9 \times 9 = 81$ $\frac{7.5 y - 3}{2} \times \frac{5 y - 8}{2} = 81$ $y^{2} - 2 y - 8 = 0$ $\Rightarrow y = 4$ $r e a c t a n g l e (5 + x) (3 + y) = 10 \times 7 = 70 ✓$
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