For the following facts are given: segment an segment ri an = 12 cm ri = 16 cm ta = 6.6 cm ni = 3.2 cm (a) assume tan is the image of tri under a dilation. is the dilation an expansion or a contraction? what is the scale factor? (b) calculate ar. show your work. (c) use the side-splitting theorem to find tn. (d) what is the ratio of the area of tri to tan ?

A) It is a contraction; the scale factor is 3/4.
B) AR = 2.2 cm.
C) TN = 9.6 cm.
D) The scale factors for the areas would be 16/9.

Explanation
A) It is a contraction because it gets smaller.  To find the scale factor, compare AN to RI:
12/16, which reduces to 3/4.

B) First we must find the length of TR.
We know that TA = 0.75TR.  We also know that TA = 6.6:

6.6 = 0.75TR

Divide both sides by 0.75:
6.6/0.75 = 0.75TR/0.75
8.8 = TR

We also know that AR = TR - TA:
AR = 8.8 - 6.6
AR = 2.2.

C) The side splitter theorem leads us to the proportion
TA/AR = TN/NI

Plugging in what we know, we have:
6.6/2.2 = TN/3.2

Cross multiply:
6.6*3.2 = 2.2TN
21.12 = 2.2TN

Divide both sides by 2.2:
21.12/2.2 = 2.2TN/2.2
9.6 = TN

D) The ratio of the area would be the square of the ratio of the sides.  The ratio of side TR to TA is 4/3; therefore the ratio of their areas would be 16/9.

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