Calculate Rectangle Area: Square with Side x Transformed by ±3cm

Question

If the length of the side of a square is X cm

(x>3)

Extend one side by 3 cm and shorten an adjacent side by 3 cm in order to obtain a rectangle.

Express the area of the rectangle using x.

Video Solution

Solution Steps

00:00 Express the area of the rectangle using X
00:03 Draw the new rectangle according to the given data
00:12 Use the formula for calculating rectangle area (side times side)
00:22 Open parentheses correctly
00:27 And this is the solution to the question

Step-by-Step Solution

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° 90\degree ) with sides of length a,b a,\hspace{4pt} b units, is given by the formula:

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

90°90°90°bbbaaabbbaaa

With this is mind, let's proceed to solve the problem:

Calculate the area of the rectangle whose vertices we'll mark with letters EFGH EFGH

We are told that one side of the rectangle is obtained by extending one side of the square with side length x x (cm) by 3 cm, and the second side of the rectangle is obtained by shortening the adjacent side of the given square by 3 cm:

x-3x-3x-3x+3x+3x+3x-3x-3x-3x+3x+3x+3HHHEEEFFFGGG

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

EF=HG=x+3EH=FG=x3 EF=HG=x+3\\ EH=FG=x-3 cm,

Apply the above formula in order to calculate the area of the rectangle that was formed from the square as described in the question:

S=EFEHS=(x+3)(x3) S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=EF\cdot EH\\ \downarrow\\ S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=(x+3)(x-3) (sq cm)

Continue to simplify the expression that we obtained for the rectangle's area, using the difference of squares formula:

(c+d)(cd)=c2d2 (c+d)(c-d)=c^2-d^2 The area of the rectangle using the above formula is as follows:

S=(x+3)(x3)S=x232S=x29 S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=(x+3)(x-3) \\ S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=x^2-3^2\\ \boxed{ S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=x^2-9} (sq cm)

Therefore, the correct answer is answer C.

Answer

x29(cm2) x^2-9\left(\operatorname{cm}²\right)