Unit 7 Dilations And Similarity Common Core Geometry Review Answers

In the realm of geometry, dilations and similarity stand as pivotal concepts, forming the bedrock for understanding transformations and relationships between figures. Mastering these concepts is crucial for students navigating Common Core Geometry. This exploration dives deep into Unit 7, offering a comprehensive review designed to solidify your understanding and prepare you to confidently tackle problems related to dilations and similarity.

Understanding Dilations: A Core Concept

At its core, a dilation is a transformation that alters the size of a figure without changing its shape. This transformation is defined by two key elements: a center of dilation and a scale factor.

• Center of Dilation: This is a fixed point from which the figure expands or contracts. Think of it as the anchor point of the transformation.
• Scale Factor: This number determines the extent of the dilation. If the scale factor is greater than 1, the figure expands (enlargement). If the scale factor is between 0 and 1, the figure shrinks (reduction). If the scale factor is 1, the figure remains the same size (congruent).

How Dilations Work: Imagine a point on the original figure. To find the corresponding point on the dilated figure, you draw a line from the center of dilation through the original point. The dilated point will lie on this same line, but its distance from the center of dilation will be the original distance multiplied by the scale factor.

Key Properties of Dilations:

• Shape is Preserved: Dilations do not change the shape of the figure. A triangle remains a triangle, a square remains a square.
• Angle Measure is Preserved: The angles within the figure stay the same. This is crucial for maintaining similarity.
• Parallel Lines Remain Parallel: If two lines are parallel in the original figure, they will remain parallel after the dilation.
• Distance is Multiplied by the Scale Factor: The distance between any two points on the figure is multiplied by the scale factor to find the corresponding distance on the dilated figure.

Exploring Similarity: Figures That Resemble

Similarity, in geometric terms, describes figures that have the same shape but potentially different sizes. This concept is inextricably linked to dilations. Two figures are similar if one can be obtained from the other through a sequence of transformations, including a dilation.

Key Characteristics of Similar Figures:

• Corresponding Angles are Congruent: This means that angles in the same position within the two figures have the same measure.
• Corresponding Sides are Proportional: The ratios of the lengths of corresponding sides are equal. This is the defining feature of similarity and is directly related to the scale factor of the dilation (if one exists).

Similarity Transformations:

To prove that two figures are similar, you need to demonstrate a series of transformations that map one figure onto the other. These transformations can include:

• Dilation: Changes the size of the figure.
• Translation: Slides the figure without rotating or reflecting it.
• Rotation: Turns the figure around a fixed point.
• Reflection: Flips the figure over a line.

Similarity Statement:

A similarity statement is a concise way to express the similarity between two figures. For example, if triangle ABC is similar to triangle XYZ, we write: ΔABC ~ ΔXYZ. The order of the vertices in the statement is crucial because it indicates which angles and sides correspond.

Common Core Geometry: Unit 7 Review - Key Concepts and Problem-Solving

Now, let's delve into some common problem types encountered in Unit 7 and explore effective strategies for solving them.

1. Identifying Dilations and Scale Factors

Problem: Given a figure and its dilated image, determine the center of dilation and the scale factor.

Solution Strategy:

1. Locate the Center of Dilation: Draw lines connecting corresponding vertices of the original figure and its image. The point where these lines intersect is the center of dilation. If the lines are parallel, the center of dilation is at infinity, and the figures are similar and have been translated.
2. Calculate the Scale Factor: Choose any pair of corresponding sides. Divide the length of the side in the dilated image by the length of the corresponding side in the original figure. This quotient is the scale factor.

Example:

Triangle ABC has vertices A(1, 1), B(2, 1), and C(1, 3). Its dilated image, triangle A'B'C', has vertices A'(2, 2), B'(4, 2), and C'(2, 6). Determine the center of dilation and the scale factor.

• Center of Dilation: By drawing lines AA', BB', and CC', we find that they intersect at the origin (0, 0). Therefore, the center of dilation is the origin.
• Scale Factor: Let's compare the lengths of side AB and A'B'. AB has a length of 1, and A'B' has a length of 2. Therefore, the scale factor is 2/1 = 2.

2. Performing Dilations on the Coordinate Plane

Problem: Given a figure and a center of dilation and a scale factor, determine the coordinates of the vertices of the dilated image.

Solution Strategy:

1. Determine the Coordinates of the Center of Dilation: Let's say the center of dilation has coordinates (h, k).
2. For Each Vertex (x, y) of the Original Figure:
   • Calculate the horizontal distance from the center of dilation: (x - h)
   • Calculate the vertical distance from the center of dilation: (y - k)
   • Multiply both distances by the scale factor, r: r(x - h) and r(y - k)
   • Add these new distances back to the coordinates of the center of dilation to find the coordinates of the corresponding vertex in the dilated image: (h + r(x - h), k + r(y - k))

Alternatively (and often more simply):

• If the center of dilation is the origin (0,0), simply multiply the coordinates of each vertex by the scale factor. (x,y) becomes (rx, ry)

Example:

Triangle DEF has vertices D(2, 3), E(4, 1), and F(1, 1). Dilate triangle DEF with a center of dilation at the origin (0, 0) and a scale factor of 3.

• D'(3 * 2, 3 * 3) = D'(6, 9)
• E'(3 * 4, 3 * 1) = E'(12, 3)
• F'(3 * 1, 3 * 1) = F'(3, 3)

3. Proving Similarity Using Transformations

Problem: Given two figures, prove that they are similar by describing a sequence of transformations that maps one figure onto the other.

Solution Strategy:

1. Analyze the Figures: Look for any obvious relationships between the figures. Are they the same shape? Are their corresponding sides proportional?
2. Identify a Possible Dilation: Determine if a dilation is necessary to change the size of one figure to match the other. Calculate the potential scale factor.
3. Determine if Other Transformations are Needed: After the dilation, will a translation, rotation, or reflection be needed to perfectly map one figure onto the other?
4. Describe the Transformations: Clearly state the sequence of transformations, including the center of dilation, scale factor, translation vector, angle of rotation, or line of reflection.

Example:

Prove that rectangle ABCD with vertices A(1, 1), B(3, 1), C(3, 2), and D(1, 2) is similar to rectangle A'B'C'D' with vertices A'(2, 2), B'(6, 2), C'(6, 4), and D'(2, 4).

• Dilation: A dilation with a center at the origin and a scale factor of 2 will transform rectangle ABCD into a rectangle with the same dimensions as A'B'C'D'.
• Description of Transformations: Rectangle ABCD can be mapped onto rectangle A'B'C'D' by a dilation with a center at the origin and a scale factor of 2.
• Conclusion: Since a dilation maps ABCD onto A'B'C'D', the two rectangles are similar.

4. Using Similarity to Solve for Unknown Side Lengths

Problem: Given two similar figures and some side lengths, find the length of an unknown side.

Solution Strategy:

1. Identify Corresponding Sides: Determine which sides of the two figures correspond to each other. The similarity statement will be helpful here.
2. Set Up a Proportion: Create a proportion using the known side lengths and the unknown side length. Remember that corresponding sides are proportional.
3. Solve for the Unknown: Solve the proportion using cross-multiplication or other algebraic techniques.

Example:

Triangle PQR is similar to triangle STU. PQ = 6, ST = 9, QR = 8, and TU = x. Find the value of x.

• Corresponding Sides: PQ corresponds to ST, and QR corresponds to TU.
• Proportion: PQ/ST = QR/TU => 6/9 = 8/x
• Solve: 6x = 72 => x = 12

5. Similarity in Right Triangles: Geometric Mean

A special case of similarity arises in right triangles when an altitude is drawn from the right angle to the hypotenuse. This altitude divides the original right triangle into two smaller right triangles that are similar to each other and to the original triangle. This leads to the concept of the geometric mean.

Theorem: If an altitude is drawn to the hypotenuse of a right triangle, then:

• The length of the altitude is the geometric mean between the lengths of the two segments of the hypotenuse.
• Each leg of the right triangle is the geometric mean between the length of the hypotenuse and the segment of the hypotenuse adjacent to that leg.

Formula:

Let the right triangle be ABC, with right angle at C. Let the altitude from C to the hypotenuse AB be CD. Let AD = x and DB = y. Then:

• CD = √(x * y)
• AC = √(x * (x + y))
• BC = √(y * (x + y))

Problem: In right triangle ABC, with right angle at C, altitude CD is drawn to hypotenuse AB. If AD = 4 and DB = 9, find the length of CD.

Solution:

CD = √(AD * DB) = √(4 * 9) = √36 = 6

6. Dilation and Parallelism: Thales' Theorem and its Converse

Thales' Theorem (also known as the Basic Proportionality Theorem): If a line is drawn parallel to one side of a triangle intersecting the other two sides, then it divides the two sides in the same ratio.

Converse of Thales' Theorem: If a line divides two sides of a triangle in the same ratio, then the line is parallel to the third side.

These theorems are directly related to dilations. Imagine dilating a triangle with the center of dilation at one of its vertices. A line connecting a point on one side to its corresponding point on the dilated side will be parallel to the opposite side of the original triangle.

Problem: In triangle ABC, point D lies on AB and point E lies on AC. If AD/DB = AE/EC, is DE parallel to BC?

Solution: Yes, by the Converse of Thales' Theorem, DE is parallel to BC.

Strategies for Success in Unit 7

• Master the Definitions: Ensure you have a solid understanding of the definitions of dilation, scale factor, similarity, and corresponding parts.
• Practice, Practice, Practice: Work through a variety of problems, including those involving coordinate geometry, proofs, and applications of similarity.
• Draw Diagrams: Visualizing the problem is often crucial. Draw accurate diagrams to help you identify corresponding sides and angles.
• Use Similarity Statements Carefully: Pay close attention to the order of vertices in similarity statements. This will help you identify corresponding parts correctly.
• Remember the Properties: Keep in mind the key properties of dilations and similar figures, such as the preservation of angle measure and the proportionality of side lengths.
• Connect Concepts: Understand how dilations and similarity are related to other geometric concepts, such as transformations, proportions, and the geometric mean.
• Review Theorems: Familiarize yourself with important theorems related to similarity, such as Thales' Theorem and its converse.

Frequently Asked Questions (FAQ)

Q: What is the difference between similarity and congruence?

A: Congruent figures are identical in both shape and size. Similar figures have the same shape but can be different sizes. Congruent figures are always similar, but similar figures are not always congruent.

Q: How can I determine the scale factor of a dilation?

A: Divide the length of a side in the dilated image by the length of the corresponding side in the original figure.

Q: Does the center of dilation affect the shape of the dilated image?

A: No, the center of dilation only affects the position of the dilated image, not its shape.

Q: Are all squares similar?

A: Yes, all squares are similar because they all have the same shape (four right angles and four congruent sides).

Q: Can a dilation have a negative scale factor?

A: Yes, a dilation can have a negative scale factor. A negative scale factor indicates that the dilated image is reflected across the center of dilation in addition to being scaled.

Conclusion: Achieving Mastery in Dilations and Similarity

Unit 7 on dilations and similarity is a cornerstone of Common Core Geometry. By understanding the fundamental concepts, mastering problem-solving strategies, and practicing consistently, you can build a strong foundation in this area. Remember to focus on the relationships between dilations and similarity, the properties of similar figures, and the applications of these concepts in various geometric problems. With dedication and perseverance, you can confidently navigate the challenges of Unit 7 and achieve mastery in dilations and similarity. The ability to visualize these transformations and apply the principles of proportionality will not only benefit you in your geometry studies but also in various real-world applications involving scaling, design, and spatial reasoning.