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1. Transformations
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United States of AmericaNC
Math 2

1.05 Transformations, congruence, and similarity

Lesson
Practice
Lesson

Congruence transformations

Using three forms of transformations, Rotations, Reflections and Translations, we can create congruent shapes.  In fact all pairs of congruent shapes can be matched to each other using a series or one or more of these three transformations.

 

Reflection (Flip)

We can reflect every point on the preimage object in a line to get our transformed image. Below is an example reflecting over the $y$y-axis. What do you notice about the coordinates of the given point when they undergo a reflection? When it is reflected over the $y$y-axis the $y$y-value stays the same, but the $x$x-value is negated.

Summary

If the point $A$A$\left(a,b\right)$(a,b) is reflected over the $y$y-axis it is transformed to the point $A'$A′$\left(-a,b\right)$(−a,b)

If the point $A$A$\left(a,b\right)$(a,b) is reflected over the $x$x-axis it is transformed to the point $A'$A′$\left(a,-b\right)$(a,−b)

 

Rotation (Turn)

A shape is rotated around a center point in a circular motion. Focus on the vertex opposite the shortest side, $\left(3,1\right)$(3,1). What happens to the coordinates when the shape is rotated $90^\circ$90° clockwise? It has rotated one quadrant, so the sign changes for one of the values (in this case the $y$y-value) and the rotation has swapped the coordinates to give us $\left(1,-3\right)$(1,−3).

Summary

If the point $A$A$\left(a,b\right)$(a,b) is rotated $90^\circ$90° clockwise about the origin it is transformed to the point $A'$A′$\left(-b,a\right)$(−b,a)

If the point $A$A$\left(a,b\right)$(a,b) is rotated $180^\circ$180° clockwise about the origin it is transformed to the point $A'$A′$\left(-a,-b\right)$(−a,−b)

If the point $A$A$\left(a,b\right)$(a,b) is rotated $270^\circ$270° clockwise about the origin it is transformed to the point $A'$A′$\left(b,-a\right)$(b,−a)

 

Translation (Slide)

The whole shape moves the same distance in the same direction, without being rotated or flipped. In the picture below, we can see the object has been moved up $5$5 units.

Summary

If the point $A$A$\left(a,b\right)$(a,b) is translated $k$k up, it is transformed to the point $A'$A′$\left(a,b+k\right)$(a,b+k)

If the point $A$A$\left(a,b\right)$(a,b) is translated $h$h units right, it is transformed to the point $A'$A′$\left(a+h,b\right)$(a+h,b)

 

 

Similarity transformations

While reflections, rotations and translations resulted in an image congruent to the preimage, dilations will result in an image which is similar to the preimage object.

Dilations (Enlargements)

We can stretch or compress every point on an object according to the same ratio to perform a dilation. Below is an example of dilating the smaller triangle by a scale factor of $2$2 from the center of enlargement $\left(1,0\right)$(1,0).

Image is similar to the preimage

Summary

For a dilation using the origin, $\left(0,0\right)$(0,0), as the center with dilation factor $a$a, the point $A$A$\left(x,y\right)$(x,y) iis transformed to the point $A'$A′$\left(ax,ay\right)$(ax,ay)

 

 

Practice questions

Question 1

Consider the figures shown.

Loading Graph...
Two triangles are depicted on a Cartesian coordinate plane with x and y-axes ranging from -10 to 10. The larger triangle, labeled with vertices A' $\left(-6,-2\right)$(−6,−2), B' $\left(2,6\right)$(2,6), and C' $\left(6,-6\right)$(6,−6), is shaded in light gray. The smaller triangle, labeled with vertices A $\left(-3,-1\right)$(−3,−1), B $\left(1,3\right)$(1,3), and C $\left(3,-3\right)$(3,−3), is shaded in dark gray and positioned inside the larger triangle.  Although they share the same shape, the triangles differ in size

  1. Are the two triangles congruent, similar or neither?

    Congruent

    A

    Similar

    B

    Neither

    C

  2. What is the transformation from triangle $ABC$ABC to triangle $A'B'C'$A′B′C′?

    Dilation

    A

    Reflection

    B

    Rotation

    C

    Translation

    D

  3. What is the scale factor of the dilation from triangle $ABC$ABC to triangle $A'B'C'$A′B′C′?

Question 2

Consider the quadrilateral with vertices at $A$A$\left(-3,-3\right)$(−3,−3), $B$B$\left(-3,3\right)$(−3,3), $C$C$\left(3,3\right)$(3,3) and $D$D$\left(3,-3\right)$(3,−3), and the quadrilateral with vertices at $A'$A′$\left(-9,-9\right)$(−9,−9), $B'$B′$\left(-9,9\right)$(−9,9), $C'$C′$\left(9,9\right)$(9,9) and $D'$D′$\left(9,-9\right)$(9,−9).

  1. Are the two rectangles similar, congruent or neither?

    congruent

    A

    similar

    B

    neither

    C

  2. What is the transformation from rectangle $ABCD$ABCD to rectangle $A'B'C'D'$A′B′C′D′?

    dilation

    A

    reflection

    B

    rotation

    C

    translation

    D

  3. What is the scale factor of the dilation of rectangle $ABCD$ABCD to rectangle $A'B'C'D'$A′B′C′D′?

Question 3

 

Outcomes

NC.M2.F-IF.1

Extend the concept of a function to include geometric transformations in the plane by recognizing that: • the domain and range of a transformation function f are sets of points in the plane; • the image of a transformation is a function of its pre-image.

NC.M2.F-IF.2

Extend the use of function notation to express the image of a geometric figure in the plane resulting from a translation, rotation by multiples of 90 degrees about the origin, reflection across an axis, or dilation as a function of its pre-image.

NC.M2.G-CO.2

Experiment with transformations in the plane. • Represent transformations in the plane. • Compare rigid motions that preserve distance and angle measure (translations, reflections, rotations) to transformations that do not preserve both distance and angle measure (e.G. Stretches, dilations). • Understand that rigid motions produce congruent figures while dilations produce similar figures.

NC.M2.G-CO.3

Given a triangle, quadrilateral, or regular polygon, describe any reflection or rotation symmetry i.E., Actions that carry the figure onto itself. Identify center and angle(s) of rotation symmetry. Identify line(s) of reflection symmetry.

NC.M2.G-CO.6

Determine whether two figures are congruent by specifying a rigid motion or sequence of rigid motions that will transform one figure onto the other.

NC.M2.G-SRT.1

Verify experimentally the properties of dilations with given center and scale factor:

NC.M2.G-SRT.1d

Dilations preserve angle measure.

NC.M2.G-SRT.2

Understand similarity in terms of transformations.

NC.M2.G-SRT.2a

Determine whether two figures are similar by specifying a sequence of transformations that will transform one figure into the other.

NC.M2.G-SRT.2b

Use the properties of dilations to show that two triangles are similar when all corresponding pairs of sides are proportional and all corresponding pairs of angles are congruent.

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