6. Describe Dilations
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A scale factor is the ratio of a length in the image to the corresponding length in the preimage.
Recall the formula for the area of a parallelogram.
1/4
The area of Q'R'S'T' is 16 times smaller than the area of QRST.
We are asked to find the scale factor for the dilation shown below. Here the image is parallelogram Q'R'S'T.
r = Length in the Image/Length in the Preimage Let's start with the preimage. We can see from the graph that the base of parallelogram QRST is 12 units. Its corresponding length in the image parallelogram Q'R'S'T' is also the base, which has a length of 3 units.
Let's substitute the values into the formula. r = Length in the Image/Length in the Preimage ⇒ r = 3/12 We can now calculate the quotient. r = 3/12 = 1/4 The scale factor for the dilation is 14.
This time we need to find the areas of the parallelograms. Once we find the areas, we will compare them.
Recall that the area of a parallelogram is its base multiplied by the height. We can find both values for each parallelogram by looking at the graph. Let's start with parallelogram QRST.
The base is 12 units and the height is 12 units. To find the area of QRST, we will multiply these values. A = ( 12)( 12) = 144 We can now move on to finding the area of Q'R'S'T'. Let's take a look at the graph one more time. This time we are looking for the base and height of parallelogram Q'R'S'T'.
The height and base length of the parallelogram are both 3units. We can multiply these values to find the area of QRST. A = (3)( 3) =9 Lastly, we need to compare the areas we found. Let's divide the area of the image, parallelogram Q'R'S'T', by the area of the preimage, parallelogram QRST. Area of the Image/Area of the Preimage ⇒ 9/144 We can calculate the quotient. 9/144 = 1/16 The area of Q'R'S'T' is 16 times smaller than the area of QRST.