Compositions of Reflections Geometry 12 4 Name the

Compositions of Reflections Geometry 12 -4

Name the Transformation Image Original Translation Review

Name the Transformation Original Image Rotation Review

Name the Transformation Original Image Reflection Review

Is this a Rigid Transformation Original Image No, it changes size Review

• Fill in the blank • The line of a reflection is the perpendicular bisector of every segment joining a point in the original figure with its image Review

• Name two types of symmetry • Reflectional • Rotational Review

The ordered pair rule (x, y) →(-x, y) is a reflection across the y-axis The ordered pair rule (x, y) →(x, -y) is a reflection across the x-axis The ordered pair rule (x, y) →(-x, -y) is a rotation about the origin The ordered pair rule (x, y) →(y, x) is a reflection across the line y = x Review

• Minimal path between points through a line • Reflect a point over the line and project a line straight to the reflected point Review

Composition of Reflections New Material

• Composition – Applying more than one transformation to a figure Vocabulary

Composition - Example

Composition - Example

Composition - Example

• Get your supplies – Patty Paper – Ruler Investigation

On a piece of patty paper, draw a small figure near one edge of the paper, and a line of reflection that does not intersect the figure Fold along the line of reflection, and trace the reflected image On your patty paper, draw a second reflection line parallel to the first so that the traced image is between the two parallel reflection lines Investigation

Investigation

How does the second traced image compare to the original figure? They are the same shape Name the single transformation that transforms the original to the second image Translation Investigation

Use a ruler to measure the distance between a point in the original figure and its second image point. Compare this distance with the distance between the parallel lines. How do the distances compare? The images are twice as far apart as the parallel lines Investigation

• Reflection across Parallel Lines Conjecture A composition of two reflections across two parallel lines is equivalent to a singe translation. In addition, the distance from any point to its second image under the two reflections is twice the distance between the parallel lines. Conjecture

Example

Example

Example

Example

• Get your supplies – Patty Paper – Protractor Each student needs one piece of patty paper Investigation

Investigation

Investigation

Investigation

Investigation

Investigation

How does the second image compare to the original figure? Same size and shape Investigation

Name the single transformation form the original to the second image Rotation Investigation

How do the angles compare? The angle of rotation is twice the angle of the intersecting lines Investigation

• Reflections across Intersecting Lines Conjecture A composition of two reflections across a pair of intersecting lines is equivalent to a single rotation. The angle of rotation is twice the acute angle between the pair of intersecting reflection lines. Conjecture

Theorem

Theorem

Example

Example

Example

Example

Example

Example

Example

Example

Example

Example

Example

• Pages 657 – 660 • 18 – 29, 63 – 65 Homework

• Pages 657 – 660 • 18 – 29, 42, 44, 63 – 65 Honors Homework