# Unit 4 Dilation and Similarity

## Lesson 1

### Learning Focus

Describe the essential features of a dilation transformation.

### Lesson Summary

In this lesson, we observed the key features of a dilation transformation while figuring out how a photocopy machine enlarges an image. We learned how to locate points on a dilated image by using the center and scale factor that define a specific dilation. We observed that “the same shape, different size” relationship between the pre-image and image figures are a consequence of the way dilations are defined.

## Lesson 2

### Learning Focus

Create similar figures by dilation given the scale factor.

Prove a theorem about the midlines of a triangle using dilations.

### Lesson Summary

In this lesson, we extended our understanding of similar figures. Since corresponding segments of similar figures are proportional, and dilations produce similar figures, corresponding parts of an image and its pre-image after a dilation are proportional. We also learned that corresponding line segments in a dilation are parallel. These two observations provided a tool for proving a theorem about the midlines of a triangle, a segment connecting the midpoints of two sides of a triangle.

## Lesson 3

### Learning Focus

Determine criteria for triangle similarity.

### Lesson Summary

In this lesson, we examined what it means to say that two figures are similar geometrically, and we examined conditions under which two triangles will be similar. We wrote and justified several theorems for triangle similarity criteria.

## Lesson 4

### Learning Focus

Prove that a line drawn parallel to one side of a triangle that intersects the other two sides divides the other two sides proportionally.

### Lesson Summary

In a previous lesson, we learned that a midline of a triangle, a line that passes through the midpoints of two of the sides, is parallel to the third side and half its length. In this lesson, we extended this theorem to include other segments that cut the sides of a triangle proportionally. We also proved a nonintuitive “side-splitting” theorem about the multiple segments formed when multiple lines parallel to a side of a triangle cut the other two sides of the triangle.

## Lesson 5

### Learning Focus

Practice using geometric reasoning in computational work.

### Lesson Summary

In this lesson, we drew upon a variety of theorems to support the computational work of finding missing sides and angles. To identify which theorems to use, we had to examine the available features of the diagram. For many measurements, multiple strategies could be used. We also used the diagram, along with our computed measurements, to develop and justify a conjecture for the sum of the interior angles of any polygon, similar to the theorem we proved previously about the sum of the interior angles in a triangle.

## Lesson 6

### Learning Focus

Prove the Pythagorean theorem algebraically.

### Lesson Summary

In today’s lesson, we learned that drawing the altitude of a right triangle from the vertex at the right angle to the hypotenuse divides the right triangle into two smaller triangles that are similar to each other and to the original right triangle. We were able to prove the Pythagorean theorem using proportionality statements about the three similar triangles.

## Lesson 7

### Learning Focus

Investigate corresponding ratios of right triangles with the same acute angle.

### Lesson Summary

In this lesson, we learned about some special ratios, called trigonometric ratios, that occur in right triangles. If two right triangles have a pair of corresponding acute angles that are congruent, the right triangles will be similar. Therefore, corresponding ratios of the sides of these two right triangles will be equal. This observation is so useful when working with right triangles that have the same acute angle that values of these ratios were recorded in tables for each acute angle between

## Lesson 8

### Learning Focus

Solve missing sides and angles in a right triangle.

Examine properties of trigonometric expressions.

### Lesson Summary

In this lesson, we examined some relationships between trigonometric ratios, such as a relationship between the sine and cosine of complementary angles. We were able to use the properties of a right triangle, including the Pythagorean theorem that describes a relationship between the lengths of the sides, to justify the observations we made today.

## Lesson 9

### Learning Focus

Solve for the missing side and angle measures in a right triangle.

### Lesson Summary

In this lesson, we extended our strategies for finding unknown sides and angles in a right triangle beyond using the Pythagorean theorem and the angle sum theorem for triangles, since sometimes we don’t have enough information in terms of side lengths or angle measures to use these theorems. We found that trigonometric ratios are useful in solving for unknown sides and that inverse trigonometric relationships are useful for finding unknown angles in a right triangle. Adding these tools allows us to find all of the missing sides and angles in a right triangle given two pieces of information: two sides of the triangle or one side and an angle.

## Lesson 10

### Learning Focus

Solve application problems using trigonometry.

### Lesson Summary

In this lesson, we learned about the modeling process and how to use right triangle trigonometry to model many different types of applications, even applications that didn’t naturally include right triangles. A right triangle became a tool for representing a situation so we could draw upon trigonometric ratios and inverse trigonometric relationships to answer important problems in construction, aviation, transportation, and other contexts.

## Lesson 11

### Learning Focus

Find missing sides of special right triangles without using trigonometry.

### Lesson Summary

In this lesson, we learned there are some special right triangles for which missing sides of the triangle can be found when only one side is known, without using trigonometry! This happens when the right triangle is the result of decomposing a familiar shape, such as a square or an equilateral triangle, into two congruent right triangles.