Question

understanding dilations and similarity
1 each problem shows three figures. two of the figures in each problem are similar, with a center of dilation at x. draw lines to determine which two figures are similar.
a.
b.
2 explain how you used lines to determine which figures are similar.
3 what is another property of similar figures?
4 jason says that all rectangles are similar because any two rectangles he chooses have 4 pairs of congruent corresponding angles. gabby says jason is wrong. who is correct? explain your answer.

Explanation:

Step1: Analyze figure a

Draw lines from center of dilation \(X\) through corresponding vertices of the figures. Figures \(A\) and \(C\) have their corresponding points lying on the same lines drawn from \(X\), so \(A\) and \(C\) are similar.

Step2: Analyze figure b

Draw lines from center of dilation \(X\) through corresponding vertices of the figures. Figures \(A\) and \(C\) have their corresponding points lying on the same lines drawn from \(X\), so \(A\) and \(C\) are similar.

Step3: Answer question 2

When lines are drawn from the center of dilation \(X\) to the corresponding vertices of the figures, if the corresponding vertices of two - figures lie on the same straight lines, then those two figures are similar. This is because in a dilation, the ratio of the distances from the center of dilation to the corresponding points of similar figures is constant.

Step4: Answer question 3

Another property of similar figures is that the ratios of their corresponding side - lengths are equal. That is, if two figures \(F_1\) and \(F_2\) are similar, and \(a_1\) and \(a_2\) are corresponding side - lengths, \(b_1\) and \(b_2\) are corresponding side - lengths, then \(\frac{a_1}{a_2}=\frac{b_1}{b_2}\).

Step5: Answer question 4

Gabby is correct. For two rectangles to be similar, not only must their corresponding angles be congruent (which is true for all rectangles since all angles in a rectangle are \(90^{\circ}\)), but the ratios of their corresponding side - lengths must also be equal. For example, a rectangle with side - lengths \(2\) and \(4\) and a rectangle with side - lengths \(3\) and \(5\) have congruent angles but are not similar because \(\frac{2}{3}
eq\frac{4}{5}\).

Answer:

1. a. Figures \(A\) and \(C\) are similar. b. Figures \(A\) and \(C\) are similar.
2. When lines are drawn from the center of dilation \(X\) to the corresponding vertices of the figures, if the corresponding vertices of two figures lie on the same straight lines, then those two figures are similar.
3. The ratios of their corresponding side - lengths are equal.
4. Gabby is correct. For two rectangles to be similar, the ratios of their corresponding side - lengths must be equal in addition to having congruent corresponding angles.