Area of the Square: Worded problems

We will refer to two separate areas: the area of the square with side y and the total area of the four squares with sides x,

We'll use the formula for the area of a square with side b:

$S=b^2$and therefore when applying it to the problem, we get that the area of the square with side y in the drawing is:

$S_1=y^2$Next, we'll calculate the area of the square with side x in the drawing:

$S_2=x^2$and to get the total area of the four squares in the drawing, we'll multiply this area by 4:

$4S_2=4x^2$Therefore, the area of the required figure in the problem, which includes the area of the square with side y and the area of the four squares with side x is:

$S_1+4S_2=y^2+4x^2$Therefore, the correct answer is A.

First, let's recall the formulas for calculating square area:

The area of a square (where all sides are equal and all angles are $90\degree$) with a side length of $a$(length units - u)

, is given by the formula:

$\boxed{ S_{\textcolor{red}{\boxed{}}}=a^2}$(square units - sq.u),

After recalling this fact, let's solve the problem:

First, let's mark the square's vertices with letters: $ABCD$

Next, considering the given data (that the square's side length is: $x+1$ cm), we'll use the above square area formula to express the area of the given square using its side length-$AB=BC=CD=DA= x +1$ (cm):

$S_{\textcolor{red}{\boxed{}}}=AB^2\\ \downarrow\\ S_{\textcolor{red}{\boxed{}}}=(x+1)^2$(sq.cm)

We'll continue and simplify the algebraic expression we got for the square's area, this will be done using the shortened multiplication formula for squaring a binomial:

$(c+d)^2=c^2+2cd+d^2$Therefore, we'll apply this formula to our square area expression:

$S_{\textcolor{red}{\boxed{}}}=(x+1)^2 \\ \downarrow\\ \boxed{S_{\textcolor{red}{\boxed{}}}=x^2+2x+1}$(sq.cm)

Therefore, the correct answer is answer D.

First, let's recall the formula for calculating the area of a rectangle:

The area of a rectangle (which has two pairs of equal opposite sides and all angles are $90\degree$) with sides of length $a,\hspace{4pt} b$ units, is given by the formula:

$\boxed{ S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=a\cdot b }$(square units)

After recalling this fact, let's solve the problem:

Let's calculate the area of the rectangle whose vertices we'll mark with letters $EFGH$ (drawing)

$$It is given in the problem that one side of the rectangle is obtained by extending one side of the square with side length $x +1$ (cm) by 1 cm, and the second side of the rectangle is obtained by shortening the adjacent side of the given square by 1 cm:

Therefore, the lengths of the rectangle's sides are:

$EF=HG=(x+1)+1\\ \downarrow\\ \boxed{ EF=HG=x+2}\\ \hspace{2pt}\\ \\ EH=FG=(x+1)-1\\ \downarrow\\ \boxed{ EH=FG=x }$ (cm)

Now we'll use the above formula to calculate the area of the rectangle that was formed from the square in the way described in the problem:

$S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=EF\cdot EH\\ \downarrow\\ S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=(x+2)x$ (sq cm)

Let's continue and simplify the expression we got for the rectangle's area, using the distributive property:

$(m+n)d=md+nd$Therefore, using the distributive property, we get that the area of the rectangle is:

$S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=(x+2)x \\ \boxed{ S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=x^2+2x}$(sq cm)

Therefore, the correct answer is answer B.