unit 7 geometry homework 13 dilations

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Homework 13 Dilations 2017

Homework 13 Dilations 2017 - Displaying top 8 worksheets found for this concept.

Some of the worksheets for this concept are Dilations date period, Unit 7 scale drawings and dilations, Geometry, Mathlinks grade 8 student packet 13 translations, Kuta geo translations, Similar triangles date period, Graph the image of the figure using the transformation, Parent and student study guide workbook.

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1. Dilations Date Period

2. unit 7 scale drawings and dilations -, 3. geometry, 4. mathlinks grade 8 student packet 13 translations ..., 5. kuta geo translations, 6. similar triangles date period, 7. graph the image of the figure using the transformation given., 8. parent and student study guide workbook.

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7.16: Dilation in the Coordinate Plane

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• Page ID 5950

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Multiplication of coordinates by a scale factor given the origin as the center.

Dilation in the Coordinate Plane

Two figures are similar if they are the same shape but not necessarily the same size. One way to create similar figures is by dilating. A dilation makes a figure larger or smaller such that the new image has the same shape as the original.

Dilation: An enlargement or reduction of a figure that preserves shape but not size. All dilations are similar to the original figure.

Dilations have a center and a scale factor . The center is the point of reference for the dilation and the scale factor tells us how much the figure stretches or shrinks. A scale factor is labeled \(k\). Only positive scale factors, \(k\), will be considered in this text.

If the dilated image is smaller than the original, then \(0<k<1\).

If the dilated image is larger than the original, then \(k>1\).

To dilate something in the coordinate plane, multiply each coordinate by the scale factor. This is called mapping . For any dilation the mapping will be \((x,y)\rightarrow (kx,ky)\). In this text, the center of dilation will always be the origin.

What if you were given the coordinates of a figure and were asked to dilate that figure by a scale factor of 2? How could you find the coordinates of the dilated figure?

For Examples 1 and 2, use the following instructions:

Given A and the scale factor, determine the coordinates of the dilated point, \(A′\). You may assume the center of dilation is the origin. Remember that the mapping will be \((x, y)\rightarrow (kx, ky)\).

Example \(\PageIndex{1}\)

\(A(−4,6), k=2\)

\(A′(−8,12)\)

Example \(\PageIndex{2}\)

\(A(9,−13), k=\dfrac{1}{2}\)

\(A′(4.5,−6.5)\)

Example \(\PageIndex{3}\)

Quadrilateral EFGH\) has vertices \(E(−4,−2)\), \(F(1,4)\), \(G(6,2)\) and \(H(0,−4)\). Draw the dilation with a scale factor of 1.5.

Remember that to dilate something in the coordinate plane, multiply each coordinate by the scale factor.

For this dilation, the mapping will be \((x,y)\rightarrow (1.5x, 1.5y) .

\(\begin{aligned} &E(−4,−2)\rightarrow (1.5(−4),1.5(−2))\rightarrow E′(−6,−3) \\ &F(1,4)\rightarrow (1.5(1),1.5(4))\rightarrow F′(1.5,6) \\ &G(6,2)\rightarrow (1.5(6),1.5(2))\rightarrow G′(9,3) \\ &H(0,−4)\rightarrow (1.5(0),1.5(−4))\rightarrow H′(0,−6)\end{aligned}\)

In the graph above, the blue quadrilateral is the original and the red image is the dilation.

Example \(\PageIndex{4}\)

Determine the coordinates of \(\Delta ABC\) and \(\Delta A′B′C′\) and find the scale factor.

The coordinates of the vertices of \(\Delta ABC\) are \(A(2,1)\), B(5,1)\) and C(3,6)\). The coordinates of the vertices of \(\Delta A′B′C′\) are A′(6,3)\), \(B′(15,3)\) and C′(9,18)\). Each of the corresponding coordinates are three times the original, so \(k=3\).

Example \(\PageIndex{5}\)

Show that dilations preserve shape by using the distance formula . Find the lengths of the sides of both triangles in Example B.

\(\begin{array}{ll} \underline{\Delta ABC} & \underline{\Delta A'B'C'} A B=\sqrt{(2-5)^{2}+(1-1)^{2}}=\sqrt{9}=3 & A^{\prime} B^{\prime}=\sqrt{(6-15)^{2}+(3-3)^{2}}=\sqrt{81}=9 \\ A C=\sqrt{(2-3)^{2}+(1-6)^{2}}=\sqrt{26} & A^{\prime} C^{\prime}=\sqrt{(6-9)^{2}+(3-18)^{2}}=3 \sqrt{26} \\ C B=\sqrt{(3-5)^{2}+(6-1)^{2}}=\sqrt{29} & C^{\prime} B^{\prime}=\sqrt{(9-15)^{2}+(18-3)^{2}}=3 \sqrt{29} \end{array} \)

From this, we also see that all the sides of \(\Delta A′B′C′\) are three times larger than \(\Delta ABC\).

Given \(A\) and \(A′\), find the scale factor. You may assume the center of dilation is the origin.

• \(A(8,2), A′(12,3)\)
• \(A(−5,−9), A′(−45,−81)\)
• \(A(22,−7), A(11,−3.5)\)

The origin is the center of dilation. Draw the dilation of each figure, given the scale factor.

• \(A(2,4), B(−3,7), C(−1,−2); k=3\)
• \(A(12,8), B(−4,−16), C(0,10); k=34\)

Multi-Step Problem Questions 6-9 build upon each other.

• Plot \(A(1,2), B(12,4), C(10,10)\). Connect to form a triangle.
• Make the origin the center of dilation. Draw 4 rays from the origin to each point from #21. Then, plot \(A′(2,4), B′(24,8), C′(20,20)\). What is the scale factor?
• Use \(k=4\), to find \(A′′B′′C′′\). Plot these points.
• What is the scale factor from \(A′B′C′\) to \(A′′B′′C′′\)?

If \(O\) is the origin, find the following lengths (using 6-9 above). Round all answers to the nearest hundredth.

• \(AA′\)
• \(AA′′\)
• \(OA′\)
• \(OA′′\)
• \(A′B′\)
• \(A′′B′′\)
• Compare the ratios \(OA:OA′\) and \(AB: A′B′\). What do you notice? Why do you think that is?
• Compare the ratios \(OA:OA′′\) and \(AB: A′′B′′\). What do you notice? Why do you think that is?

Review (Answers)

To see the Review answers, open this PDF file and look for section 7.12.

Additional Resources

Interactive Element

Video: Dilation in the Coordinate Plane Principles - Basic

Activities: Dilation in the Coordinate Plane Discussion Questions

Study Aids: Types of Transformations Study Guide

Practice: Dilation in the Coordinate Plane

Free Printable Dilations Worksheets for 7th Grade

Dilations: Discover a collection of free printable math worksheets tailored for Grade 7 students, designed to help educators effectively teach and assess understanding of math dilations concepts.

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Explore printable Dilations worksheets for 7th Grade

Dilations worksheets for Grade 7 are an essential resource for teachers looking to help their students master the concepts of geometry and transformations in math. These worksheets provide a variety of engaging and challenging exercises that focus on understanding the properties of dilations, including scale factors, center of dilation, and the effect on the size and position of figures. By incorporating these worksheets into their lesson plans, teachers can ensure that their Grade 7 students develop a strong foundation in geometry, which is crucial for success in higher-level math courses. Additionally, these worksheets can be used as a tool for formative assessment, allowing teachers to monitor their students' progress and identify areas where additional support may be needed. Dilations worksheets for Grade 7 are a valuable addition to any math teacher's toolkit.

Quizizz is an excellent platform for teachers to complement their use of dilations worksheets for Grade 7 and expand their students' learning experience. With Quizizz, teachers can create interactive quizzes and games that cover a wide range of topics, including math, geometry, and transformations. This platform allows teachers to track their students' progress and provide immediate feedback, making it an effective tool for both formative and summative assessments. In addition to quizzes, Quizizz also offers a variety of other resources, such as flashcards and interactive lessons, which can be used to reinforce and extend the concepts covered in the dilations worksheets. By incorporating Quizizz into their teaching strategies, Grade 7 math teachers can provide a more engaging and dynamic learning environment that supports their students' success in geometry and beyond.

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Unit 10: Transformations

About this unit.

In this topic you will learn about the most useful math concept for creating video game graphics: geometric transformations, specifically translations, rotations, reflections, and dilations.

You will learn how to perform the transformations, and how to map one figure into another using these transformations.

Introduction to rigid transformations

• Rigid transformations intro (Opens a modal)
• Translations intro (Opens a modal)
• Rotations intro (Opens a modal)
• Identify transformations 4 questions Practice

Translations

• Translating shapes (Opens a modal)
• Determining translations (Opens a modal)
• Translation challenge problem (Opens a modal)
• Properties of translations (Opens a modal)
• Translations review (Opens a modal)
• Translate points 4 questions Practice
• Determine translations 4 questions Practice
• Translate shapes 4 questions Practice
• Rotating shapes (Opens a modal)
• Determining rotations (Opens a modal)
• Rotating shapes about the origin by multiples of 90° (Opens a modal)
• Rotations review (Opens a modal)
• Rotating shapes: center ≠ (0,0) (Opens a modal)
• Rotate points 4 questions Practice
• Determine rotations 4 questions Practice
• Rotate shapes 4 questions Practice
• Rotate shapes: center ≠ (0,0) 4 questions Practice

Reflections

• Reflecting shapes: diagonal line of reflection (Opens a modal)
• Determining reflections (advanced) (Opens a modal)
• Reflecting shapes (Opens a modal)
• Reflections review (Opens a modal)
• Reflect points 4 questions Practice
• Determine reflections 4 questions Practice
• Determine reflections (advanced) 4 questions Practice
• Reflect shapes 4 questions Practice
• Advanced reflections 4 questions Practice

Rigid transformations overview

• No videos or articles available in this lesson
• Find measures using rigid transformations 4 questions Practice
• Rigid transformations: preserved properties 4 questions Practice
• Mapping shapes 4 questions Practice
• Performing dilations (Opens a modal)
• Dilating shapes: shrinking by 1/2 (Opens a modal)
• Dilating shapes: expanding (Opens a modal)
• Dilate points 4 questions Practice
• Dilations: scale factor 4 questions Practice
• Dilations: center 4 questions Practice
• Dilate triangles 4 questions Practice
• Dilations and properties 4 questions Practice

Properties and definitions of transformations

• Precisely defining rotations (Opens a modal)
• Identifying type of transformation (Opens a modal)
• Sequences of transformations 4 questions Practice
• Defining transformations 4 questions Practice
• Intro to reflective symmetry (Opens a modal)
• Intro to rotational symmetry (Opens a modal)
• Finding a quadrilateral from its symmetries (Opens a modal)
• Finding a quadrilateral from its symmetries (example 2) (Opens a modal)
• Reflective symmetry of 2D shapes 4 questions Practice

Old transformations videos

• Performing translations (old) (Opens a modal)
• Performing rotations (old) (Opens a modal)
• Performing reflections: rectangle (old) (Opens a modal)
• Performing reflections: line (old) (Opens a modal)
• Determining translations (old) (Opens a modal)
• Rotation examples (old) (Opens a modal)
• Determining rotations (old) (Opens a modal)
• Dilating lines (Opens a modal)