Question

look at this parallelogram: if both dimensions are tripled, then which of the following statements about its perimeter will be true? the ratio of the new perimeter to the old perimeter will be 4:1. the ratio of the new perimeter to the old perimeter will be 2:1.

Explanation:

Step1: Calculate the old perimeter

Let the sides of the parallelogram be \(a = 3\) in and \(b = 6\) in. The perimeter formula for a parallelogram is \(P=2(a + b)\). So, \(P_{old}=2(3 + 6)=2\times9 = 18\) in.

Step2: Calculate the new perimeter

If both dimensions are tripled, \(a_{new}=3\times3 = 9\) in and \(b_{new}=3\times6 = 18\) in. Then \(P_{new}=2(9 + 18)=2\times27 = 54\) in.

Step3: Calculate the ratio

The ratio of the new perimeter to the old perimeter is \(\frac{P_{new}}{P_{old}}=\frac{54}{18}=3:1\). But wait, let's use the property of perimeter of similar - figures. For a parallelogram (a type of quadrilateral), if the scale factor of side - lengths is \(k\) (here \(k = 3\)), the perimeter of a polygon \(P=2(a + b)\), and if \(a'=ka\) and \(b'=kb\), then \(P'=2(ka+kb)=k\times2(a + b)\). So the ratio of perimeters of two similar parallelograms (same shape, different sizes) is equal to the ratio of their corresponding side - lengths.

Since the original statements in the problem are incorrect (maybe there was a mis - type in the problem's options, but if we assume the general formula), if we use the formula \(P = 2(a + b)\) and \(P'=2(3a+3b)=3\times2(a + b)\)

Answer:

None of the given options (The ratio of the new perimeter to the old perimeter will be \(3:1\))