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Math  /  Geometry

QuestionLearn with an example ✓\checkmark
In the graph below, △I′J′K′\triangle I^{\prime} J^{\prime} K^{\prime} is the image of △IJK\triangle I J K after a dilation.
What are the scale factor and center of the dilation? Simplify your answers and write them as fractions or whole numbers. scale factor: □\square center of the dilation: ( □\square , □\square ) Submit Work it out

Studdy Solution

STEP 1

1. The dilation transforms △IJK\triangle IJK into △I′J′K′\triangle I'J'K'.
2. The scale factor is determined by comparing corresponding sides or distances from the center of dilation.
3. The center of dilation can be found by determining the point that remains invariant under the dilation.

STEP 2

1. Determine the scale factor of the dilation.
2. Identify the center of the dilation.

STEP 3

To determine the scale factor, compare the distances between corresponding points of △IJK\triangle IJK and △I′J′K′\triangle I'J'K'.
- Calculate the distance between corresponding points II and I′I':
Distance from origin to I=(2−0)2+(4−0)2=4+16=20 \text{Distance from origin to } I = \sqrt{(2-0)^2 + (4-0)^2} = \sqrt{4 + 16} = \sqrt{20}
Distance from origin to I′=(6−0)2+(12−0)2=36+144=180 \text{Distance from origin to } I' = \sqrt{(6-0)^2 + (12-0)^2} = \sqrt{36 + 144} = \sqrt{180}
- The scale factor kk is given by:
k=Distance to I′Distance to I=18020=9×2020=32020=3 k = \frac{\text{Distance to } I'}{\text{Distance to } I} = \frac{\sqrt{180}}{\sqrt{20}} = \frac{\sqrt{9 \times 20}}{\sqrt{20}} = \frac{3\sqrt{20}}{\sqrt{20}} = 3

STEP 4

To find the center of dilation, observe that the center is the point from which all corresponding points are equidistantly scaled.
- Since the dilation is uniform, the center of dilation can be found by solving the equations of the lines passing through corresponding points. For simplicity, observe that the origin (0,0)(0,0) is a common center for such transformations when the scale factor is consistent across all points.
- Check if the origin (0,0)(0,0) is the center by verifying that the lines II′II', JJ′JJ', and KK′KK' intersect at the origin.
The scale factor is:
3 \boxed{3}
The center of the dilation is:
(0,0) \boxed{(0, 0)}

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