In Module 2, Topic A students revisit what scale drawings are and discover two systematic methods of how to create them using dilations. The comparison of the two methods yields the Triangle Side Splitter Theorem and the Dilation Theorem.

View Standards
**Standard(s): **
[MA2015] GEO (9-12) 10 :

[MA2015] GEO (9-12) 16 :

[MA2015] GEO (9-12) 17 :

[MA2019] GEO-19 (9-12) 32 :

10 ) Prove theorems about triangles. *Theorems include measures of interior angles of a triangle sum to 180*^{o}, base angles of isosceles triangles are congruent, the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length, and the medians of a triangle meet at a point. [G-CO10]

[MA2015] GEO (9-12) 16 :

16 ) Use the properties of similarity transformations to establish the angle-angle (AA) criterion for two triangles to be similar. [G-SRT3]

[MA2015] GEO (9-12) 17 :

17 ) Prove theorems about triangles. *Theorems include a line parallel to one side of a triangle divides the other two proportionally, and conversely; and the Pythagorean Theorem proved using triangle similarity.* [G-SRT4]

[MA2019] GEO-19 (9-12) 32 :

32. Use coordinates to prove simple geometric theorems algebraically.

[MA2019] GEO-19 (9-12) 34 : 34. Use congruence and similarity criteria for triangles to solve problems in real-world contexts.

This math brainteaser challenges you to find a simple, elegant solution to a seemingly complex problem! Students will use geometry principles and their knowledge about triangles to solve this puzzle. Can you figure it out?

View Standards
**Standard(s): **
[MA2015] GEO (9-12) 17 :

[MA2019] GEO-19 (9-12) 26 :

17 ) Prove theorems about triangles. *Theorems include a line parallel to one side of a triangle divides the other two proportionally, and conversely; and the Pythagorean Theorem proved using triangle similarity.* [G-SRT4]

[MA2019] GEO-19 (9-12) 26 :

26. Verify experimentally the properties of dilations given by a center and a scale factor.

a. Verify that a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.

b. Verify that the dilation of a line segment is longer or shorter in the ratio given by the scale factor.

[MA2019] GEO-19 (9-12) 38 : 38. Use the mathematical modeling cycle involving geometric methods to solve design problems.

*Examples: Design an object or structure to satisfy physical constraints or minimize cost; work with typographic grid systems based on ratios; apply concepts of density based on area and volume.*

View Standards
**Standard(s): **
[MA2015] GEO (9-12) 17 :

[MA2019] GEO-19 (9-12) 26 :

17 ) Prove theorems about triangles. *Theorems include a line parallel to one side of a triangle divides the other two proportionally, and conversely; and the Pythagorean Theorem proved using triangle similarity.* [G-SRT4]

[MA2019] GEO-19 (9-12) 26 :

26. Verify experimentally the properties of dilations given by a center and a scale factor.

a. Verify that a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.

b. Verify that the dilation of a line segment is longer or shorter in the ratio given by the scale factor.

Module 2, Topic B is an in-depth study of the properties of dilations. Though students applied dilations in Topic A, their use in the ratio and parallel methods was to establish relationships that were consequences of applying a dilation, not directly about the dilation itself. In Topic B, students explore observed properties of dilations (Grade 8 Module 3) and reason why these properties are true. This reasoning is possible because of what students have studied regarding scale drawings and the triangle side-splitter and dilation theorems. With these theorems, it is possible to establish why dilations map segments to segments, lines to lines, etc. Some of the arguments involve an examination of several sub-cases; it is in these instances of thorough examination that students must truly make sense of problems and persevere in solving them (MP.1).

In Lesson 6, students revisit the study of rigid motions and contrast the behavior of the rigid motions to that of a dilation. Students confirm why the properties of dilations are true in Lessons 7–9. Students repeatedly encounter G.SRT.A.1a and b in these lessons and build arguments with the help of the ratio and parallel methods (G.SRT.B.4). In Lesson 10, students study how dilations can be used to divide a segment into equal divisions. Finally, in Lesson 11, students observe how the images of dilations of a given figure by the same scale factor are related, as well as the effect of a composition of dilations on the scale factor of the composition.

View Standards
**Standard(s): **
[MA2015] GEO (9-12) 16 :

[MA2015] GEO (9-12) 17 :17 ) Prove theorems about triangles. *Theorems include a line parallel to one side of a triangle divides the other two proportionally, and conversely; and the Pythagorean Theorem proved using triangle similarity.* [G-SRT4]

[MA2019] GEO-19 (9-12) 27 :

16 ) Use the properties of similarity transformations to establish the angle-angle (AA) criterion for two triangles to be similar. [G-SRT3]

[MA2015] GEO (9-12) 17 :

[MA2019] GEO-19 (9-12) 27 :

27. Given two figures, determine whether they are similar by identifying a similarity transformation (sequence of rigid motions and dilations) that maps one figure to the other.

[MA2019] GEO-19 (9-12) 34 : 34. Use congruence and similarity criteria for triangles to solve problems in real-world contexts.

[MA2019] GEO-19 (9-12) 36 : 36. Use geometric shapes, their measures, and their properties to model objects and use those models to solve problems.

Students learn what it means for two figures to be similar in general, and then focus on triangles and what criteria predict that two triangles will be similar. Length relationships within and between figures are studied closely and foreshadows work in Module 2, Topic D. The topic closes with a look at how similarity has been used in real-world application.

View Standards
**Standard(s): **
[MA2015] GEO (9-12) 17 : 17 ) Prove theorems about triangles. *Theorems include a line parallel to one side of a triangle divides the other two proportionally, and conversely; and the Pythagorean Theorem proved using triangle similarity.* [G-SRT4]

In Module 2, Topic D, students use their understanding of similarity and focus on right triangles as a lead-up to trigonometry. In Lesson 21, students use the AA criterion to show how an altitude drawn from the vertex of the right angle of a right triangle to the hypotenuse creates two right triangles similar to the original right triangle. Students examine how the ratios within the three similar right triangles can be used to find unknown side lengths. Work with lengths in right triangles lends itself to expressions with radicals. In Lessons 22 and 23, students learn to rationalize fractions with radical expressions in the denominator and also to simplify, add, and subtract radical expressions. In the final lesson of Topic D, students use the relationships created by an altitude to the hypotenuse of a right triangle to prove the Pythagorean theorem.