Unit 4: Dilation and Similarity
Lessons
Focus Standards
Learning Focus
Additional Resources
A Develop Understanding Task
Develops understanding about dilation and the essential features of a dilation. Surfaces the need for a center of dilation and a scale factor when describing a dilation.
Describe the essential features of a dilation transformation.
How do I use a dilation to enlarge or shrink a figure?
How are distance and area in a dilated image related to corresponding distances and area in the original image?
A Solidify Understanding Task
Solidifies the definition of dilation and how to perform one. Examines proportionality relationships in triangles that are known to be similar to each other based on dilations, including the theorem that a line parallel to one side of a triangle divides the other two proportionally.
Create similar figures by dilation given the scale factor.
Prove a theorem about the midlines of a triangle using dilations.
How do I know if two geometric figures are similar?
How do I know if two geometric figures are similar by dilation?
What interesting characteristics of an image are produced by dilating a polygon centered at one of the vertices of the polygon?
A Solidify Understanding Task
Compares two definitions of similarity for polygons based on dilations. Provides an opportunity to prove AA, SAS, and SSS similarity theorems for triangles.
Determine criteria for triangle similarity.
What is the difference between the common usage of the word similar (e.g., rectangles are more similar to each other than rectangles and triangles are), and the mathematical conventions for the word? What does it mean for two polygons to be similar?
How can I prove (or disprove) that two triangles are similar?
A Solidify Understanding Task
Examines proportionality relationships of segments when two sides of a triangle are crossed by a line that is parallel to the third side.
NC.M2.G-SRT.4
Prove that a line drawn parallel to one side of a triangle that intersects the other two sides divides the other two sides proportionally.
What observations can I make about the segments formed on two sides of a triangle by a line drawn parallel to the third side?
What if the parallel line is below the base?
How might I justify my observations algebraically?
A Practice Understanding Task
Practices applying theorems about lines, angles, and proportional relationships to find missing angles and lengths when parallel lines are crossed by multiple transversals. Develops proof for the sum of interior angles for an n-gon.
Practice using geometric reasoning in computational work.
How do I look for structure in a diagram so I can use familiar features of the diagram to find the measures of unknown sides and angles?
What measurements do I need to calculate first, in order to calculate additional measurements?
A Practice Understanding Task
Uses similar triangles to prove the Pythagorean theorem and by extension, theorems about geometric means in right triangles. Practices writing proportionality statements about similar triangles.
Prove the Pythagorean theorem algebraically.
When and how do I use algebra in a geometric proof?
What does each proof of the Pythagorean theorem reveal?
A Develop Understanding Task
Uses understanding of similar triangles to develop an understanding of right triangle trigonometric relationships (sine, cosine, tangent).
Investigate corresponding ratios of right triangles with the same acute angle.
What determines if two right triangles are similar?
Why are the ratios of sides in right triangles so special that they deserve a classification of their own (trigonometric ratios)?
Lesson 8: Relationships with Meaning
A Solidify Understanding Task
Solidifies understanding of trigonometric ratios. Examines relationships between the sine and cosine ratios for right triangles, including the Pythagorean identity.
Solve missing sides and angles in a right triangle.
Examine properties of trigonometric expressions.
When do I use the Pythagorean theorem and when do I use trigonometry when solving for missing parts of a right triangle?
What observations can I make about the relationships between trigonometric ratios of the two different reference angles in a right triangle?
How do the properties of a right triangle influence algebraic statements involving trigonometric expressions?
Lesson 9: Finding the Value of a Relationship
A Solidify Understanding Task
Solidifies methods for finding unknown values of right triangles, including Pythagorean theorem and trigonometric ratios. Develops understanding of using inverse trigonometric relationships to find unknown angle measures.
NC.M2.G-SRT.8
Solve for the missing side and angle measures in a right triangle.
While we can find missing side lengths and angles in a right triangle if two sides or two angles are known using the Pythagorean theorem or the angle sum theorem for triangles, what information do we need to know to find missing measures in a right triangle now that trigonometric ratios are available as a computational tool?
How do we use trigonometry to make indirect measurements when the object can’t be measured directly?
Lesson 10: Using Trigonometric Relationships
A Practice Understanding Task
Practices setting up and solving right triangles to model real-world contexts.
NC.M2.G-SRT.8
Solve application problems using trigonometry.
How do I apply trigonometric ratios to practical problems?
What are the essential elements of modeling a real-world context using a right triangle, even when only an imaginary right triangle exists?
A Solidify Understanding Task
Reviews and builds on prior knowledge of right triangle trigonometry by examining the relationship of sides in special right triangles.
NC.M2.G-SRT.12
Find missing sides of special right triangles without using trigonometry.
Why are 30-60-90 degrees and 45-45-90 degree triangles considered to be “special?”
Why can we find the missing sides of these right triangles without using trigonometry?
Are there any other special right triangles?