Area Comparison: Square vs. Modified Rectangle with Side X cm

Question

The side length of a square is X cm

$(x>3)$

We extend one side by 3 cm and shorten an adjacent side by 3 cm, and we get a rectangle.

Which shape has a larger area?

00:00 Which square has a larger area?

00:03 Let's draw the square, its sides are greater than 3

00:08 Let's draw the new rectangle according to the given data

00:24 We added and subtracted 3 from the appropriate sides according to the given data

00:31 We'll use the formula for calculating the area of a square (side squared)

00:34 We'll substitute appropriate values and solve to find the area

00:37 Let's calculate the rectangle's area (length times width)

00:40 We'll substitute appropriate values and solve to find the rectangle's area

00:47 Let's make sure to use parentheses properly

00:53 The rectangle's area equals the square's area minus 9, therefore it's smaller

00:57 And that's the solution to the problem

To determine which shape has a larger area, we need to compare the areas of the square and the rectangle:

The side length of the square is $X$ , so its area is given by:

\text{Area of square} = X^2

The dimensions of the rectangle are $X + 3$ cm and $X - 3$ cm. Thus, its area is:

\text{Area of rectangle} = (X + 3)(X - 3)

Using the difference of squares formula, we find:

(X + 3)(X - 3) = X^2 - 9

We compute the difference between the square's area and the rectangle's area:

X^2 - (X^2 - 9) = 9

Since 9 is positive, the area of the square is larger than the area of the rectangle.

Therefore, the square has a larger area than the rectangle.

The square