D AY 71 – V ERIFYING DILATIONS
I NTRODUCTION We have already encountered dilations in our previous lessons on transformations. A dilation produces an image that is the same shape as the pre-image but same or different size, therefore, it is worth noting that dilation is not a rigid transformation in general. In this lesson, we will base our discussion on the effects of dilations on line segments as part of verifying dilations. We will discover what happens to a line segment when dilation is performed with the center of the dilation on the segment and with the center at another point, not on the segment. We will also relate the dilation of a line segment to the ratio given by the scale factor.
V OCABULARY 1. Dilation A transformation that enlarges or reduces the pre- image while preserving shape by moving all points along a ray passing through a fixed point called the center of dilation. 2. Center of dilation A fixed point in the plane where all the points on the pre-image are moved along a ray passing through this point. 3. Scale factor of the dilation The ratio of the length of corresponding sides on the image and the pre-image
4. Collinear points Points that lie on the same straight line.
Before we verify the effects of dilations on line segments, it is important to recall some key properties of dilations: 1. The pre-image and the image have the same shape but different sizes, unless the scale factor is equal to 1 2. Angles are mapped to congruent angles 3. Parallel lines are mapped to parallel lines 4. The ratios of corresponding sides on the pre- image and the image are equal.
5. The ratios of corresponding line segments are equal to the scale factor 6. Line segments are mapped to line segments with a given scale factor
We can now discuss the following concepts about dilations and lines and line segments: 1. A dilation leaves a line passing through the center of dilation unchanged, that is, the image lies on the same line. If an entire line is dilated with the center of dilation on the line, the line remains unchanged. After a dilation, image points and corresponding pre-image points all lie on the straight line to the center of dilation. This concept applies to line segments on plane figures too.
Consider line AB (AB) shown below. Note that the center of dilation, O is on AB . If a dilation with scale factor 2 is performed on points A and B, they will be mapped to points A′ and B ′ respectively. Note that the points A,B, A′ and B ′ are collinear. A O B A′ B′ If we dilate the entire AB about point O, by using point A as the pre-image point then point A′ should lie on the line through point O and point A which still lies on AB . This means that since point O is located on AB then the images of all points on AB will lie on AB .
We can therefore conclude that, if a line is dilated in such a way that the center of dilation lies on the line, the dilation does not change the line, that is, we get the same line. Note that in case of line segments, the pre-image and the image both line on the same line. A O B A′ B′ In the figure above, AB and its image A ′ B′ both lie on the same line, AB after a dilation with scale factor 2 about the point O.
2. A dilation takes a line not passing through the center of the dilation to a parallel line We can also dilate a line when the center of dilation does not lie on the line. Consider the diagram below which shows a dilation of PQ with point O as the center of dilation and a scale factor of 2. Note that the point O is not on PQ. The points O, P and P′ are collinear. Similarly, the points O, Q and Q ′ are collinear.
Dilations preserve angle measures, therefore, ∠OPQ ≅ ∠OP ′ Q′ . P P′ O Q Q′
If a pair of lines is intersected by a transversal and the resulting corresponding angles are congruent, then the two lines are parallel, thus PQ ∥ P ′ Q′ . Based on the illustration above, we can conclude that a dilation takes a line not passing through the center of the dilation to a parallel line. In case of line segments on plane figures, the line segments are mapped to parallel line segments. In the figure above, PQ and its image P ′ Q′ are parallel.
3. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. If the scale factor, 𝑙 is greater than 1 : (a) the resulting image is larger than the pre- image. (b) the line segments, which are the sides of the image will be longer than the corresponding sides of the pre-image. (c) it is referred to as an enlargement. (d) the pre-image lies between the center of dilation and the image.
This illustrated below using ΔXYZ and its image ΔX′Y′Z′ after a dilation about the point O, scale factor, 𝑙 > 1 . X′ X O Z Y Y′ Z′
According to the figure above; X ′ Y′ > XY , Y ′ 𝑎′ > YZ and X ′ Z′ > XZ If we use a scale factor greater than 1, say, 3, we shall have: X ′ Y′ ≅ 3XY , Y ′ 𝑎′ ≅ 3YZ and X ′ Z′ ≅ 3XZ This shows that the lengths of the line segments on the image will be three times longer than those of the corresponding segments on the pre-image. The scale factor is taken as the ratio of the length of any side on the image to the length of the corresponding side on the object, that is X ′ Y′ XY = Y ′ Z′ YZ = X ′ Z′ XZ
If the scale factor lies between 0 and 1 : (a) the resulting image is smaller than the pre- image. (b) the line segments, which are the sides of the image will be shorter than the corresponding sides of the pre-image. (c) it is referred to as an reduction. (d) The image lies between the center of dilation and the pre-image. If the scale factor is equal to 1 , the pre-image and the image are congruent
This illustrated below using ΔXYZ and its image ΔX′Y′Z′ after a dilation about the point O, scale factor 𝑙 such that 0 < 𝑙 < 1 . X X′ O Z′ Y′ Z Y
According to the figure above; X ′ Y′ < XY , Y ′ 𝑎′ < YZ and X ′ Z′ < XZ If we use a scale factor 𝑙 such that, 0 < 𝑙 < 1 , say, 1 4 , we shall have: 1 1 1 4 XY , Y ′ 𝑎′ ≅ 4 YZ and X ′ Z′ ≅ X ′ Y′ ≅ 4 XZ
Example If the length of one side on a triangle is 4.4 inches, what would be the length of the corresponding side 3 after the triangle is dilated by a scale factor of 2 about a point O located away from the triangle? State whether the side and its corresponding side on the image will be parallel or not.
Solution Length of side on the image Scale factor = Length of corresponding side on the object Length of side on the image = Scale factor × Length of corresponding side on the object = 3 2 × 4.4 = 6.6 𝑗𝑜𝑑ℎ𝑓𝑡 The side and its corresponding side on the image will be parallel.
HOMEWORK Use a ruler to dilate AB about the center C with a scale factor of 2. Estimate the locations of points A and B on the line. State whether A′B′ lies on AB or not. A C B
A NSWERS TO HOMEWORK A′B′ lies on AB . A C B A′ B′
THE END
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