The Area of EFGH is 2x+9 which is similar to The Area of JKLM is 2x-6. The perimeter of EFGH is x+3 which is similar to the Perimeter of JKLM is x-1. FInd all Possible values of X.

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]