A square living room in a home has an area of 110.25 square feet. What is the side length of the living room?

a = 10.5 feet

Step-by-step explanation:

Given that,

The area of a square room, A = 110.25 ft²

We need to find the side length of the living room. Let the side be x.

We know that,

The area of a square,

So,

So, the side length of the living room is equal to 10.5 feet.

step 1: draw a parallelogram slanting right and label the base 12 and the slant height 8.

step 2: draw a diagonal from top left to bottom right.

step 3: construct an altitude from the top left vertex down to the base. call that "h".

step 4: call the left side segment "a" and the right side segment "b". a + b = 12.

step 5: we have 2 pythagorean theorems: (i) a%5E2+%2B+h%5E2+=+8%5E2; (ii) b%5E2+%2B+h%5E2+=+10%5E2.

step 6: by subtracting the two, we get (iii) b%5E2+-+a%5E2+=+36.

step 7: find values for a and b such that (b+a)(b-a) = 36. If b= 9 and a = 3, we have the values.

By using either (i) or (ii), this also gives us the height, h, as sqrt(55).

step 8: draw the other diagonal. Since we are dealing with a parallelogram, opposite sides have the same slope. So, the base grows from 12 to 15.

step 9: we apply another pythagorean theorem to our new triangle. (iv) 15%5E2+%2B+sqrt%2855%29%5E2+=+d%5E2

step 10: solve for d. we get d^2 = 225 + 55 = 280. So, d ~ 16.7332.

Hope this helps :)