Question

the area of a square is 400 square units. select all the statements that would be true if the length of each side of the square increased by one

the area of the square would be a rational number.

the area of the square would be an irrational number.

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each side length would be a perfect square.

the area of the square would be a perfect square.

the area of the square would be a nonterminating, nonrepeating decimal.

Explanation:

1. First, find the original side length of the square. The area of a square is given by \( A = s^2 \), where \( s \) is the side length. Given \( A = 400 \), we solve \( s^2 = 400 \), so \( s=\sqrt{400} = 20 \) units.
2. If each side length is increased by some amount (assuming it's a positive real number, but likely an integer or rational number in context), let's say the new side length is \( s'=20 + x \) (where \( x \) is the increase). The new area \( A'=(20 + x)^2=400 + 40x+x^2 \).
3. - Rational Number Check: If \( x \) is a rational number (e.g., if we increase by 1, \( x = 1 \), new side is 21, area is \( 21^2=441 \); if we increase by 0.5, \( x = 0.5 \), new side is 20.5, area is \( 20.5^2 = 420.25\)), the area will be a rational number because the sum and product of rational numbers are rational. So "The area of the square would be a rational number" is true.

• Irrational Number Check: Since the area is a sum and product of rational numbers (if \( x \) is rational) or can be expressed as a fraction (if \( x \) is a real number, but in the context of square side length increase, it's likely a rational change), the area won't be irrational. So this statement is false.
• Side Length as Perfect Square Check: The new side length is \( 20 + x \). A perfect square is a number like \( 1,4,9,16,25,\dots \). Unless \( x \) is chosen specifically (e.g., \( x = 5 \) to make side 25, a perfect square), in general, the side length won't be a perfect square. So this statement is false.
• Perfect Square Area Check: The new area \( A'=(20 + x)^2 \), which is a perfect square by definition (since it's the square of \( (20 + x) \)). So "The area of the square would be a perfect square" is true.
• Non - terminating, Non - repeating Decimal Check: Since the area is a perfect square (a rational number, either an integer or a terminating decimal if \( x \) is a decimal with finite digits), it won't be a non - terminating, non - repeating decimal. So this statement is false.

Answer:

• The area of the square would be a rational number.
• The area of the square would be a perfect square.