Bellwork Find the geometric mean of 5 & 18 Find the geometric mean of 5 & 18 Find the geometric mean of 3 & 44 Find the geometric mean of 3 & 44 Solve for x Solve for x Solve for x, if AB & CD are parallel Solve for x, if AB & CD are parallel What point is twice as far from the origin as (3, 5)? What point is twice as far from the origin as (3, 5)? x x A B C D x 2x No Clickers

Bellwork Solution Find the geometric mean of 5 & 18 Find the geometric mean of 5 & 18

Bellwork Solution Find the geometric mean of 3 & 44 Find the geometric mean of 3 & 44

Bellwork Solution Solve for x Solve for x x x-2 5 3

Bellwork Solution Solve for x, if AB & CD are parallel Solve for x, if AB & CD are parallel A B C D x 2x

Bellwork Solution What point is twice as far from the origin as (3, 5)? What point is twice as far from the origin as (3, 5)?

Perform Similiarity Transformations Section 6.7

Test on Wednesday

The Concept We’ve covered most of chapter 6, but we have yet to apply our understanding of similarity to objects on the coordinate plane. We’ve covered most of chapter 6, but we have yet to apply our understanding of similarity to objects on the coordinate plane. Today we’re going to use our understanding of similarity and transformations to discuss similarity transformations. Today we’re going to use our understanding of similarity and transformations to discuss similarity transformations.

Review We’ve seen three kinds of transformations thus far We’ve seen three kinds of transformations thus far Translations Translations Shifts or moves up or down and right or left Shifts or moves up or down and right or left Rotations Rotations Object rotations a direction and angle about the origin Object rotations a direction and angle about the origin Reflections Reflections Flips of an object either over the x-axis or the y-axis Flips of an object either over the x-axis or the y-axis The last one that we will learn about is dilations The last one that we will learn about is dilations

Definitions Dilation Dilation Special kind of transformation that stretches or shrinks a figure to create a similar figure Special kind of transformation that stretches or shrinks a figure to create a similar figure Figures are either reduced or enlarged Figures are either reduced or enlarged Type of similarity transformation Type of similarity transformation

Definitions Center of Dilation Center of Dilation Fixed point with which the object is dilated Fixed point with which the object is dilated Scale factor of dilation Scale factor of dilation Ratio of a side length of the image to the corresponding side length of the original figure Ratio of a side length of the image to the corresponding side length of the original figure

Coordinate Notation We prefer to be able to notate for dilations We prefer to be able to notate for dilations For dilations centered at the origin For dilations centered at the origin (x,y)  (kx,ky), where k is a scale factor (x,y)  (kx,ky), where k is a scale factor If 0<k<1, reduction If 0<k<1, reduction If k>1, enlargement If k>1, enlargement

Drawing a Dilation Draw a dilation of an object with vertices (0,2), (5, 3) & (5,-3) using a scale factor of 2 Draw a dilation of an object with vertices (0,2), (5, 3) & (5,-3) using a scale factor of 2

Drawing a Dilation Draw a dilation of an object with vertices (4,6), (2, 4) & (6,-6) using a scale factor of 1/2 Draw a dilation of an object with vertices (4,6), (2, 4) & (6,-6) using a scale factor of 1/2

Example Draw a dilation of a quadrilateral ABCD with vertices A(2,2), B(4,2), C(4,0), D(0,-2). Use a scale factor of 1.5 and label the object FGHJ

Scale or k factor We’ve discussed scale factor before and defined it as We’ve discussed scale factor before and defined it as The quotient of a side length of the second object and the corresponding side length of the first object The quotient of a side length of the second object and the corresponding side length of the first object This property of dilations is no different This property of dilations is no different For example, find the scale factor of these two objects For example, find the scale factor of these two objects 1 2

Example We can also determine k factor from points We can also determine k factor from points Find the k factor between these two objects Find the k factor between these two objects What do we need in order to give an accurate answer

Example Is the green object a dilation of the yellow one? Is the green object a dilation of the yellow one? How do we know?

Practical Example You are using a photo quality printer to enlarge a digital picture. The picture on the computer screen is 6 centimeters by 6 centimeters. The printed image is 15 cm by 15 cm. What is the scale factor of the enlargement? You are using a photo quality printer to enlarge a digital picture. The picture on the computer screen is 6 centimeters by 6 centimeters. The printed image is 15 cm by 15 cm. What is the scale factor of the enlargement?

Homework 6.7 Exercises 6.7 Exercises 1, 2-8 even, 9-23, 25, 26 1, 2-8 even, 9-23, 25, 26

Example Draw a dilation of a quadrilateral ABCD with vertices A(-3,5), B(3,4), C(4,-2), D(-3,-2). Use a scale factor of 1.5 and label the object FGHJ

Most Important Points Definition of Dilation Definition of Dilation Bounds for the k scalar Bounds for the k scalar Performing Dilations Performing Dilations Finding k factor from points Finding k factor from points