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DAY ERIFYING DILATIONS - hi · PDF file 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

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  • DAY 71 – VERIFYING DILATIONS

  • INTRODUCTION

    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.

  • VOCABULARY

    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.

    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.

    OA B B′A′

  • 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.

    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.

    OA B B′A′

  • 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′.

    O P

    Q

    P′

    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, 𝑘 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, 𝑘 > 1.

    X

    Y Z

    X′

    Y′ Z′

    O

  • According to the figure above;

    X′Y′ > XY, Y′𝑍′ > YZ and X′Z′ > XZ

    If we use a scale factor greater than 1, say, 3, we

    shall have:

    X′Y′ ≅ 3XY, Y′𝑍′ ≅ 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 𝑘 such that 0 < 𝑘 < 1.

    X

    Y Z

    X′

    Y′ Z′

    O

  • According to the figure above;

    X′Y′ < XY, Y′𝑍′ < YZ and X′Z′ < XZ

    If we use a scale factor 𝑘 such that, 0 < 𝑘 < 1, say, 1

    4 , we shall have:

    X′Y′ ≅ 1

    4 XY, Y′𝑍′ ≅

    1

    4 YZ and X′Z′ ≅

    1

    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

    after the triangle is dilated by a scale factor of 3

    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

    Scale factor = Length of side on the image

    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 𝑖𝑛𝑐ℎ𝑒𝑠

    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.

    CA B

  • ANSWERS TO HOMEWORK

    A′B′ lies on AB.

    CA B B′A′

  • THE END