Respuesta :
Answer:
[tex]x = 9[/tex]
Step-by-step explanation:
Assume EFGH and JKLM are rectangles
JKLM appears to be the smaller of both.
So:
[tex]EF = n * JK[/tex] and [tex]GH = n * LM[/tex]
Where n is the scale of dilation
The area of EFGH is:
[tex]Area = EF * GH[/tex] ----- (1)
Substitute [tex]EF = n * JK[/tex] and [tex]GH = n * LM[/tex]
[tex]Area = n*JK * n*LM[/tex]
[tex]Area = n^2*JK * LM[/tex]
So, we have:
[tex]Area = EF * GH[/tex] and [tex]Area = n^2*JK * LM[/tex]
[tex]EF * GH = 2x + 9[/tex] --- Given
[tex]JK * LM = 2x - 6[/tex] --- Given
So:
[tex]Area = EF * GH[/tex] and [tex]Area = n^2*JK * LM[/tex]
[tex]Area = 2x + 9[/tex] and [tex]Area = n^2 * (2x - 6)[/tex]
Equate bot areas
[tex]2x + 9 = n^2*(2x - 6)[/tex]
Make n^2 the subject
[tex]n^2 = \frac{2x + 9}{2x - 6}[/tex]
The perimeter (P) of EFGH is:
[tex]P = 2*(EF + GH)[/tex] ----- (1)
Substitute [tex]EF = n * JK[/tex] and [tex]GH = n * LM[/tex]
[tex]P = 2*(n*JK + n*LM)[/tex]
[tex]P = 2n*(JK + LM)[/tex]
So, we have:
[tex]P = 2*(EF + GH)[/tex] and [tex]P = 2n*(JK + LM)[/tex]
[tex]2*(EF + GH) = x + 3[/tex] --- Given
[tex]2*(JK + LM) = x - 1[/tex] --- Given
So:
[tex]P = 2*(EF + GH)[/tex] and [tex]P = 2n*(JK + LM)[/tex]
[tex]P =x + 3[/tex] and [tex]P = n *(x - 1)[/tex]
Equate bot perimeters
[tex]x + 3 = n(x - 1)[/tex]
Make n the subject
[tex]n =\frac{x + 3}{x - 1}[/tex]
Square both sides
[tex]n^2 =\frac{(x + 3)^2}{(x - 1)^2}[/tex]
Recall that: [tex]n^2 = \frac{2x + 9}{2x - 6}[/tex]
So, we have:
[tex]\frac{(x + 3)^2}{(x - 1)^2} = \frac{2x + 9}{2x - 6}[/tex]
Evaluate all squares
[tex]\frac{x^2 + 6x +9}{x^2 - 2x + 1} = \frac{2x + 9}{2x - 6}[/tex]
Cross multiply
[tex](x^2 + 6x +9)(2x - 6) = (x^2 - 2x + 1)(2x + 9)[/tex]
Open brackets
[tex]2x^3 +12x^2 + 18x - 6x^2 -36x -54 = 2x^3 - 4x^2 + 2x + 9x^2 - 18x +9[/tex]
Subtract 2x^3 from both sides
[tex]12x^2 + 18x - 6x^2 -36x -54 = - 4x^2 + 2x + 9x^2 - 18x +9[/tex]
Collect like terms
[tex]12x^2 - 6x^2 + 18x -36x -54 = - 4x^2 + 9x^2+ 2x - 18x +9[/tex]
[tex]6x^2 - 18x -54 = 5x^2- 16x +9[/tex]
Collect like terms
[tex]6x^2 -5x^2 - 18x +16x -54 -9= 0[/tex]
[tex]x^2 -2x -63= 0[/tex]
Expand
[tex]x^2 +7x - 9x - 63 = 0[/tex]
Factorize
[tex]x(x +7) - 9(x + 7) = 0[/tex]
[tex](x -9) (x + 7) = 0[/tex]
[tex]x = 9\ or\ -7[/tex]
x can not be negative.
So: [tex]x = 9[/tex]