R is a 4 by x rectangle. M is a 7 by 11-x rectangle. Determine the value of x that make R and M similar

R is 4 by x
M is 7 by (11-x)

we need these to be similar

R and M similar means the dimensions are proportional
so

..x...........11-x
▬▬..=..▬▬▬
...4...........7

We solve this proportion by noting that
cross products of a true proportion are equal.

Since a product is an answer to a multiplication
problem, we perform cross multiplications and
set them equal. Thus

7x = 4(11-x)

use distributive law on right hand side

7x = 44 - 4x

add 4x to both sides of this equation to
obtain

11x = 44

consequently, by dividing both sides by 11
we find that

........44
x = ▬▬ which implies x=4
........11

Consequently x=4 IS a correct answer despite your assertion in
the additional details. Now there may be another correct answer.

Here is how. We now let 11 - x and 4 be the corresponding
sides and then the x and the 7 are corresponding. Now we
solve the proportion

..x.............7
▬▬..=..▬▬▬
...4..........11 -x

This is different from the way it is worded in the problem but would still produce
two similar rectangles.

Now
28 = x(11- x)

thus

28 = 11x - x²

add x² - 11x to both sides and we obtain

x² - 11x + 28 = 0

that factors to

(x-4)(x-7) =0

so x=4 or x=7 are the possible solutions.

Now we have identified 4 as a possible solution.
The other solution is x=7 .

If we use x=4 then

R is 4 by 4 and M is 7 by 7
which not only means they are similar but
the rectangles are in fact, squares.

If we use x=7, then

R is 4 by 7 and M is 7 by 4
and the two are just rectangles, but they
are not only similar, THEY are CONGRUENT.

Rectangles are similar then then sides are in proportion
4/7 = x/(11 – x)
4(11 – x) = 7x
44 – 4x = 7x
11x = 44
x = 4
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