Congruent Squares Question using algebra

Question

I am also struggling with this question where I don't know how to use my knowledge of congruency to work out the length of the longer side.

I thought the longest side would be a+x but it was wrong. How would I work out the length for this square.

Question:

Answer 1

I just want to say that I'm assuming that x is the side length of the inner square even though it looks like another line was highlighted.

Let's call the smaller length of one rectangle w and the longer length l

So here are the formulas we know:

a=l+w

For x we can see that two upright triangles on either side of the square span out to length a therefore:

x=a-2w

Now all we need to do is solve the equation below by substituting a for its formula:

x=(l+w)- 2w

x+2w=l+w

x-w=l

However. This is not over. Because we have to do this in terms of x and a. We cannot use w. We can simply re-arrange the formula for x (x=a-2w) to do that.

w=(a-x)/2

Now we replace w in the 'final' equation with its formula:

x-((a-x)/2)=l