Question

6.) triangle abc is dilated to get triangle abc
a.) which of these scale factors for the dilation would result in an image that was smaller than the original figure? circle all the apply.
a.) $\frac{3}{5}$ b.) $\frac{13}{17}$ c.) 1 d.) $\frac{4}{3}$
b.) explain your selection(s) in part (a).

Explanation:

Step1: Recall the rule of dilation scale - factor

If the scale factor \(k\) of a dilation satisfies \(0 < k<1\), the image is smaller than the original figure. If \(k = 1\), the image is congruent to the original figure. If \(k>1\), the image is larger than the original figure.

Step2: Analyze each option

• For option A: \(\frac{3}{5}=0.6\), and \(0<0.6 < 1\).
• For option B: \(\frac{13}{17}\approx0.765\), and \(0 < 0.765<1\).
• For option C: \(k = 1\), the image is congruent to the original.
• For option D: \(\frac{4}{3}\approx1.33\), and \(1.33>1\).

for part (b):
When the scale factor \(k\) of a dilation is between \(0\) and \(1\) (\(0 < k<1\)), each side length of the image triangle \(A'B'C'\) is \(k\) times the corresponding side length of the original triangle \(ABC\). Since \(k=\frac{3}{5}=0.6\) and \(k = \frac{13}{17}\approx0.765\) are both in the interval \((0,1)\), the side lengths of the dilated triangles (with these scale - factors) are shorter than the side lengths of the original triangle \(ABC\). So the images (triangles) are smaller than the original figure.

Answer:

A. \(\frac{3}{5}\), B. \(\frac{13}{17}\)