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Lesson 10Composing Figures

Let’s use reasoning about rigid transformations to find measurements without measuring.

Learning Targets:

  • I can find missing side lengths or angle measures using properties of rigid transformations.

10.1 Angles of an Isosceles Triangle

Here is a triangle.

A triangle labeled A B C with horizontal side B C labeled 2 and sides A B and A C are each labeled 3.
  1. Reflect triangle ABC over line  AB . Label the image of C  as C’ .
  2. Rotate triangle ABC’ around A so that C’ matches up with B .

  3. What can you say about the measures of angles B and C ?

10.2 Triangle Plus One

Here is triangle ABC .

Triangle ABC is shown.

  1. Draw midpoint M of side AC .

  2. Rotate triangle ABC 180 degrees using center M to form triangle CDA . Draw and label this triangle.

  3. What kind of quadrilateral is ABCD ? Explain how you know.

Are you ready for more?

In the activity, we made a parallelogram by taking a triangle and its image under a 180-degree rotation around the midpoint of a side. This picture helps you justify a well-known formula for the area of a triangle. What is the formula and how does the figure help justify it?

10.3 Triangle Plus Two

The picture shows 3 triangles. Triangle 2 and Triangle 3 are images of Triangle 1 under rigid transformations.

3 congruent triangles are shown forming a trapezoid.

  1. Describe a rigid transformation that takes Triangle 1 to Triangle 2. What points in Triangle 2 correspond to points A , B , and C in the original triangle?

  2. Describe a rigid transformation that takes Triangle 1 to Triangle 3. What points in Triangle 3 correspond to points A , B , and C in the original triangle?

  3. Find two pairs of line segments in the diagram that are the same length, and explain how you know they are the same length.

  4. Find two pairs of angles in the diagram that have the same measure, and explain how you know they have the same measure.

10.4 Triangle ONE Plus

Here is isosceles triangle ONE . Its sides ON and OE have equal lengths. Angle O is 30 degrees. The length of ON is 5 units.

An isosceles triangle is shown with two side lengths of 5 and one angle of 30 degrees.

  1. Reflect triangle ONE across segment ON . Label the new vertex M .

  2. What is the measure of angle MON ?

  3. What is the measure of angle MOE ?

  4. Reflect triangle MON across segment OM . Label the point that corresponds to N as T .

  5. How long is \overline{OT} ? How do you know?

  6. What is the measure of angle TOE ?

  7. If you continue to reflect each new triangle this way to make a pattern, what will the pattern look like?

Lesson 10 Summary

Earlier, we learned that if we apply a sequence of rigid transformations to a figure, then corresponding sides have equal length and corresponding angles have equal measure. These facts let us figure out things without having to measure them!

For example, here is triangle ABC .

A triangle A, B, C where the interior angle at A has measure 36 degrees.

We can reflect triangle ABC across side AC to form a new triangle:

Triangle A, B, C, with angle with measure 36 degrees at A. It has been reflected in the side A, B.

Because points A and C are on the line of reflection, they do not move. So the image of triangle ABC is AB'C . We also know that:

  • Angle B'AC measures 36^\circ because it is the image of angle BAC .
  • Segment AB' has the same length as segment AB .

When we construct figures using copies of a figure made with rigid transformations, we know that the measures of the images of segments and angles will be equal to the measures of the original segments and angles.

Lesson 10 Practice Problems

  1. Here is the design for the flag of Trinidad and Tobago.

    The flage of Trinidad and Tobago is shown.
    “The Flag of Trinidad and Tobago” via Wikimedia Commons. Public Domain.

    Describe a sequence of translations, rotations, and reflections that take the lower left triangle to the upper right triangle.

  2. Here is a picture of an older version of the flag of Great Britain. There is a rigid transformation that takes Triangle 1 to Triangle 2, another that takes Triangle 1 to Triangle 3, and another that takes Triangle 1 to Triangle 4.

    An image of an older version of the flag of Great Britain. The flag is a rectangle with a vertical length about twice the width. Red stripes divide the flag in half vertically and horizontally. White stripes connect the vertices along diagonals, crossing behind the red stripes. The remaining area is composed of 8 blue right triangles. At the top of the flag, 2 large right triangles line up on either side of the vertical red stripe by their shorter square sides, so that they are mirror images of each other. At the bottom of the flag, 2 large right triangles line up on either side of the vertical red stripe by their shorter square sides, so that they are mirror images of each other. At the left side, 2 small right triangles line up on either side of the horizontal red stripe by their longer square sides so that they are mirror images of each other. The triangle above the red stripe is labeled 1; the triangle below the red strip is labeled 3. At the right side, 2 small right triangles line up on either side of the horizontal red stripe by their longer square sides so that they are mirror images of each other. The triangle above the red stripe is labeled 2; the triangle below the red strip is labeled 4.
    “Flag of Great Britain (1707–1800)” by Hoshi via Wikimedia Commons. Public Domain.
    1. Measure the lengths of the sides in Triangles 1 and 2. What do you notice?
    2. What are the side lengths of Triangle 3? Explain how you know.
    3. Do all eight triangles in the flag have the same area? Explain how you know.
    1. Which of the lines in the picture is parallel to line \ell ? Explain how you know.
      Four lines are drawn so that one line, labeled “p”, intersects the other 3 lines, which are labeled “m,” “k,” and “l.” Lines “k” and “l” will not intersect no matter how far they extend. Both are perpendicular to line “p.” Line “m” is not perpendicular “p” and appears to be angled towards “k” and “l” so that it would intersect them at a point not shown.
    2. Explain how to translate, rotate or reflect line \ell to obtain line k .
    3. Explain how to translate, rotate or reflect line \ell to obtain line p .

  4. Point A has coordinates (3,4) . After a translation 4 units left, a reflection across the x -axis, and a translation 2 units down, what are the coordinates of the image?

  5. Here is triangle XYZ :

    Traingle XYZ is shown

    Draw these three rotations of triangle XYZ together.

    1. Rotate triangle XYZ 90 degrees clockwise around Z .
    2. Rotate triangle XYZ 180 degrees around Z .
    3. Rotate triangle XYZ 270 degrees clockwise around Z .