Class Greeting

Chapter 7 – Lesson 1 Dilation

Objective: The students will be able to apply the concepts involving Dilation.

Vocabulary dilation scale factor enlargement reduction isometry dilation center of dilation

A dilation is a transformation that changes the size of a figure but not its shape. The preimage and the image are always similar.

A scale factor describes how much the figure is enlarged or reduced A scale factor describes how much the figure is enlarged or reduced. For a dilation with scale factor k, you can find the image of a point by multiplying each coordinate by k: (a, b)  (ka, kb).

If the scale factor of a dilation is greater than 1 (k > 1), it is an enlargement. If the scale factor of a dilation is a positive number less than 1 (k < 1), it is a reduction. Helpful Hint Helpful Hint If the scale factor of a dilation is 1 (k = 1), it is an isometry dilation. It produces an image that coincides with the preimage. The two figures are congruent and on top of each other.

A dilation with center O.

Example 1: Drawing Dilations Draw the image of ∆WXYZ under a dilation with a scale factor of 2 and the center of dilation P. Step 1 Draw a line through P and each vertex. Step 2 On each line, mark twice the distance from P to the vertex. W’ X’ Step 3 Connect the vertices of the image. Y’ Z’

Step 1 Draw a line through Q and each vertex. Your Turn Copy the figure and the center of dilation. Draw the dilation of RSTU using center Q and a scale factor of 3. Step 1 Draw a line through Q and each vertex. Step 2 On each line, mark twice the distance from Q to the vertex. R’ S’ Step 3 Connect the vertices of the image. T’ U’

Example 2: Dilations On a sketch of a flower, 4 in. represent 1 in. on the actual flower. If the flower in the sketch has a 3 in. diameter, find the diameter of the actual flower. The scale factor in the dilation is 4. Let the diameter of the actual flower be d in.       4d = 3 d = 0.75 in. Answer: the diameter of the actual flower is 0.75 in.

Your Turn A rectangle on a transparency has length 6cm and width 4 cm . On the transparency 1 cm represents 12 cm on the projection. Find the perimeter of the rectangle in the projection.             Answer: the perimeter of the rectangle in the projection is 240 cm.

Example 3: Drawing Dilations in the Coordinate Plane Graph RSTU and its images after a composition of dilations centered at the origin with a scale factor of and a scale factor of 2, given R(0, 0), S(4, 0), T(2, -2), and U(–2, –2). The first dilation of (x, y) is

Graph the preimage and images. Preimage RSTU: R(0, 0), S(4, 0), T(2, -2), and U(–2, –2) T’’ U’’ R’ S’ T’ U’ R S T U R’’ S’’

Your Turn Draw the image of the triangle with vertices P(–4, 4), Q(–2, –2), and R(4, 0) under a dilation with a scale factor of centered at the origin. The dilation of (x, y) is

Graph the preimage and image. Q’ R’ R Q

Example 4: Drawing Dilations in the Coordinate Plane   Draw rays from A through the vertices of RST. Graph point A and RST   R S R’ T S’ T’ A

Draw rays from A through the vertices of RST. Your Turn Graph RST and its image after a dilation centered at A(20, 14) with a scale factor of 3 given R(14, 13), S(19, 11) and T(14, 10). Draw rays from A through the vertices of RST. Graph point A and RST Multiply the distances from A to the vertices of RST by the scale factor 3 and plot those points. A shortcut is to multiply the “ups” and “overs” by 3. A R R’ S T S’ T’

Kahoot!

Lesson Summary: Objective: The students will be able to apply the concepts involving Dilation.

Preview of the Next Lesson: Objective: The students will solve problems involving Similar Polygons.

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