Question

look at this square: 1 in 1 in if the side lengths are quadrupled, then which of the following statements about its perimeter will be true? the new perimeter will be 2 times the old perimeter. the new perimeter will be \\( \frac { 1 } { 2 } \\) of the old perimeter.

Explanation:

Step1: Calculate the old perimeter

The formula for the perimeter of a square is \(P = 4s\), where \(s\) is the side - length. Given \(s = 1\) in, the old perimeter \(P_{old}=4\times1 = 4\) in.

Step2: Calculate the new side - length and new perimeter

If the side - length is quadrupled, the new side - length \(s_{new}=4\times1 = 4\) in. Then the new perimeter \(P_{new}=4\times s_{new}=4\times4 = 16\) in.

Step3: Find the relationship between new and old perimeters

We want to find \(\frac{P_{new}}{P_{old}}\). \(\frac{P_{new}}{P_{old}}=\frac{16}{4}=4\). But let's check the general formula. If the original side - length is \(s\), \(P_{old}=4s\). If the new side - length is \(ks\) (here \(k = 4\)), then \(P_{new}=4(ks)=k(4s)\). So when \(k = 4\), \(P_{new}=4P_{old}\). Wait, maybe there was a mis - understanding of the problem. Let's re - examine.

Wait, no, let's use another approach. Let the original side be \(a\), perimeter \(P_1=4a\). New side \(b = 4a\), new perimeter \(P_2=4b=4\times(4a)=16a\). But if we assume the problem was mis - written (maybe double instead of quadruple, but working with the given options).

Wait, no, let's check the options:
Let the original side \(s\), perimeter \(P = 4s\). If we assume a wrong approach (maybe the user made a typo in'side - length change description). Let's assume the problem was about doubling (but no). Wait, no, formula: perimeter of square \(P = 4s\). If \(s\) becomes \(4s\), \(P\) becomes \(4\times(4s)=16s\). But if we check the options:
Let’s assume the original perimeter \(P_1 = 4s\), new perimeter \(P_2=4\times(4s)\). But if we use the formula \(P = 4s\), and if we consider the ratio.

Alternatively, let's use a property: For a square (a regular polygon), if the side - length is scaled by a factor \(k\), the perimeter is scaled by the same factor \(k\). Here, if the side - length is quadrupled (\(k = 4\)), perimeter is quadrupled. But since the options are wrong (maybe a mis - print in the problem). Wait, no, let's check again.

Wait, original perimeter \(P_1=4\times1 = 4\). If side is quadrupled (new side \(4\)), new perimeter \(P_2=4\times4 = 16\). But if we assume the problem was about doubling (side becomes \(2\), perimeter \(8\), which is \(2\) times the original \(4\)).

Answer:

The new perimeter will be 4 times the old perimeter. But since the options are wrong (maybe a mis - print in the problem statement. If we assume the intended scaling factor was 2 (double instead of quadruple), then the new perimeter will be 2 times the old perimeter.