Question

3.29. raksha is puzzled. she is working with the parallelogram drawn at right and wants to make it smaller instead of bigger.
a. what should she do if she wants the sides of her new figure to be half as long as the original sides? what zoom factor should she use? find the dimensions of her new figure.
b. while drawing some other shapes, raksha ended up with a shape congruent to the original parallelogram. what is the common ratio between pairs of corresponding sides?

Explanation:

Step1: Determine the zoom factor

Since the new figure is half as long as the original, the zoom factor \(k\) is calculated by the ratio of the new length to the original length. If we assume the original side lengths are \(a = 16\) and \(b=11\), and the new side lengths are \(a'=\frac{a}{2}\) and \(b'=\frac{b}{2}\). The formula for the zoom factor \(k=\frac{\text{new length}}{\text{original length}}\). So \(k = \frac{1}{2}\)

Step2: Calculate new dimensions

For a side of length \(l\) in the original parallelogram, the new side length \(l'\) is given by \(l'=k\times l\).
If \(l = 16\), then \(l'=\frac{1}{2}\times16 = 8\)
If \(l = 11\), then \(l'=\frac{1}{2}\times11=5.5\)

Step3: Find the common ratio for congruent shapes

For congruent shapes (same - size and same - shape), the ratio of corresponding sides is \(1\) because \(k = 1\) (a non - scaled or \(100\%\) scaled figure)

Answer:

a. Zoom factor \(k=\frac{1}{2}\), new dimensions: \(8\) and \(5.5\)
b. The common ratio between pairs of corresponding sides for congruent shapes is \(1\)