Question

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.

Answer 1

`x = 9`

Explanation:

Assume EFGH and JKLM are rectangles

JKLM appears to be the smaller of both.

So:

`EF = n * JK` and
`GH = n * LM`

Where n is the scale of dilation

The area of EFGH is:

`Area = EF * GH` ----- (1)

Substitute
`EF = n * JK` and
`GH = n * LM`

`Area = n*JK * n*LM`

`Area = n^2*JK * LM`

So, we have:

`Area = EF * GH` and
`Area = n^2*JK * LM`

`EF * GH = 2x + 9` --- Given

`JK * LM = 2x - 6` --- Given

So:

`Area = EF * GH` and
`Area = n^2*JK * LM`

`Area = 2x + 9` and
`Area = n^2 * (2x - 6)`

Equate bot areas

`2x + 9 = n^2*(2x - 6)`

Make n^2 the subject

`n^2 = (2x + 9)/(2x - 6)`

The perimeter (P) of EFGH is:

`P = 2*(EF + GH)` ----- (1)

Substitute
`EF = n * JK` and
`GH = n * LM`

`P = 2*(n*JK + n*LM)`

`P = 2n*(JK + LM)`

So, we have:

`P = 2*(EF + GH)` and
`P = 2n*(JK + LM)`

`2*(EF + GH) = x + 3` --- Given

`2*(JK + LM) = x - 1` --- Given

So:

`P = 2*(EF + GH)` and
`P = 2n*(JK + LM)`

`P =x + 3` and
`P = n *(x - 1)`

Equate bot perimeters

`x + 3 = n(x - 1)`

Make n the subject

`n =(x + 3)/(x - 1)`

Square both sides

`n^2 =((x + 3)^2)/((x - 1)^2)`

Recall that:
`n^2 = (2x + 9)/(2x - 6)`

So, we have:

`((x + 3)^2)/((x - 1)^2) = (2x + 9)/(2x - 6)`

Evaluate all squares

`(x^2 + 6x +9)/(x^2 - 2x + 1) = (2x + 9)/(2x - 6)`

Cross multiply

`(x^2 + 6x +9)(2x - 6) = (x^2 - 2x + 1)(2x + 9)`

Open brackets

`2x^3 +12x^2 + 18x - 6x^2 -36x -54 = 2x^3 - 4x^2 + 2x + 9x^2 - 18x +9`

Subtract 2x^3 from both sides

`12x^2 + 18x - 6x^2 -36x -54 = - 4x^2 + 2x + 9x^2 - 18x +9`

Collect like terms

`12x^2 - 6x^2 + 18x -36x -54 = - 4x^2 + 9x^2+ 2x - 18x +9`

`6x^2 - 18x -54 = 5x^2- 16x +9`

Collect like terms

`6x^2 -5x^2 - 18x +16x -54 -9= 0`

`x^2 -2x -63= 0`

Expand

`x^2 +7x - 9x - 63 = 0`

Factorize

`x(x +7) - 9(x + 7) = 0`

`(x -9) (x + 7) = 0`

`x = 9\ or\ -7`

x can not be negative.

So:
`x = 9`