Video: Using Properties of Dilations or Enlargements to Identify a Scale Factor

The quadrilateral 𝐴𝐡𝐢𝐷 in the given figure has been dilated from the center point π‘₯ to the quadrilateral 𝐴′𝐡′𝐢′𝐷′. What is the scale factor of the dilation?

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Video Transcript

The quadrilateral 𝐴𝐡𝐢𝐷 in the given figure has been dilated from the center point π‘₯ to the quadrilateral 𝐴 dash 𝐡 dash 𝐢 dash 𝐷 dash. What is the scale factor of the dilation?

So, in this question, we’re given a four-sided shape, 𝐴𝐡𝐢𝐷. The image of 𝐴𝐡𝐢𝐷 is this smaller quadrilateral shown. And we’re told it’s dilated from the center point π‘₯. Now, normally, to find the scale factor of the dilation, we can use a formula. We find two corresponding sides, one on each of our shapes. The scale factor is the length of the side on the image divided by the length of the side on the original shape. And so, in this case, for example, we could take the length of the side 𝐴𝐷 and the length of the side 𝐴 dash 𝐷 dash.

If we take one small square to be one unit, then 𝐴 dash 𝐷 dash is 10 units in length. Similarly, the side length 𝐴𝐷 is 20 units. So, we could say that the scale factor of enlargement was 10 divided by 20, which is equal to one-half. In this case, though, that’s not the end of the story. If we join rays connecting 𝐴 and 𝐴 dash and 𝐷 and 𝐷 dash, they pass through the center point π‘₯. And we should recall that an enlargement using a negative scale factor achieves this. It causes the enlargement to appear on the other side of the center, and it’s also inverted as our shapes are.

So, in this case, the scale factor of enlargement or the scale factor of dilation is actually negative one-half.

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