LESSON 9–6 Dilations

Five-Minute Check (over Lesson 9–5) TEKS Then/Now Key Concept: Dilation Example 1: Draw a Dilation Example 2: Real-World Example: Find the Scale of a Dilation Key Concept: Dilations in the Coordinate Plane Example 3: Dilations in the Coordinate Plane Example 4: Compositions of Dilations Lesson Menu

D. This figure does not have line symmetry. State whether the figure appears to have line symmetry. If so, how many lines of symmetry does it have? A. yes; 4 lines B. yes; 3 lines C. yes; 2 lines D. This figure does not have line symmetry. 5-Minute Check 1

This figure does not have line symmetry. State whether the figure appears to have line symmetry. If so, how many lines of symmetry does it have? A. yes; 8 lines B. yes; 4 lines yes; 2 lines This figure does not have line symmetry. 5-Minute Check 2

The figure has rotational symmetry The figure has rotational symmetry. State the order and magnitude of symmetry. A. 5; 72° B. 5; 45° C. 6; 60° D. 6; 72° 5-Minute Check 3

The figure has rotational symmetry The figure has rotational symmetry. State the order and magnitude of symmetry. A. 8; 60° B. 8; 45° C. 10; 45° D. 10; 36° 5-Minute Check 4

What is the order and magnitude of symmetry of a regular hexagon? A. order 2, magnitude 180° B. order 3, magnitude 120° C. order 6, magnitude 60° D. order 12, magnitude 30° 5-Minute Check 5

Mathematical Processes G.1(C), G.1(E) Targeted TEKS G.3(A) Describe and perform transformations of figures in a plane using coordinate notation. G.3(B) Determine the image or pre-image of a given two-dimensional figure under a composition of rigid transformations, a composition of non-rigid transformations, and a composition of both, including dilations where the center can be any point in the plane. Also addresses G.10(B). Mathematical Processes G.1(C), G.1(E) TEKS

Draw dilations in the coordinate plane. You identified dilations and verified them as similarity transformations. Draw dilations. Draw dilations in the coordinate plane. Then/Now

Concept

Since k > 1, the dilation is an enlargement of trapezoid PQRS. Draw a Dilation Copy trapezoid PQRS and point C. Then use a ruler to draw the image of trapezoid PQRS under a dilation with center C and scale factor 3. Since k > 1, the dilation is an enlargement of trapezoid PQRS. Example 1

Draw a Dilation Copy trapezoid PQRS and point C. Then use a ruler to draw the image of trapezoid PQRS under a dilation with center C and scale factor 3. Locate P', Q', R', and S' so that Example 1

Answer: Draw trapezoid P'Q'R'S'. Draw a Dilation Copy trapezoid PQRS and point C. Then use a ruler to draw the image of trapezoid PQRS under a dilation with center C and scale factor 3. Answer: Draw trapezoid P'Q'R'S'. Example 1

Which diagram shows the dilation image of ΔLMN with center C and ? A. B. C. D. Example 1

Find the Scale Factor of a Dilation PUPPETS To create the illusion of a “life-sized” image, puppeteers sometimes use a light source to show an enlarged image of a puppet projected on a screen or wall. Suppose that the distance between a light source L and the puppet is 24 inches (LP). To what distance PP' should you place the puppet from the screen to create a 49.5-inch tall shadow (I'M') from a 9-inch puppet? Example 2

Find the Scale Factor of a Dilation Analyze This problem involves a dilation. The center of the dilation is L, LP = 24 in., IM = 9 in., I'M' = 49.5 in. You are asked to find PP'. Formulate Find the scale factor of the dilation from the preimage IM to the image I'M'. Use the scale factor to find LP and then use LP and LP' to find PP'. Example 2

Find the Scale Factor of a Dilation Determine The scale factor k of the enlargement is the ratio of the length on the image to a corresponding length on the preimage. Scale factor of image image = I'M', preimage = IM Divide. Example 2

Use this scale factor of 5.5 to find LP'. Find the Scale Factor of a Dilation Use this scale factor of 5.5 to find LP'. LP' = k(LP) Definition of dilation = 5.5(24) k = 5.5 and LP = 24 = 132 Multiply. Use LP' and LP to find PP'. LP + PP' = LP' Segment Addition 24 + PP' = 132 LP = 24 and LP' = 132 PP' = 108 Subtract 24 from each side. Example 2

Evaluate Since 5.5 > 1, the scale factor is reasonable. Find the Scale Factor of a Dilation Answer: So, the puppet should be placed so that the distance from it to the screen (PP') is 108 inches. Justify Since the dilation is an enlargement, the scale factor should be greater than 1. Evaluate Since 5.5 > 1, the scale factor is reasonable. Example 2

PUPPETS Suppose you have a similar situation with the puppet and light source. The distance between the light source L and the puppet is 30 inches (LP). To what distance should you place the puppet from the screen to create a 54-inch tall shadow (I'M') from a 6-inch puppet? A. 100 inches B. 180 inches C. 220 inches D. 240 inches Example 2

Concept

Multiply the x- and y-coordinates of each vertex by the scale factor, Dilations in the Coordinate Plane A. Trapezoid EFGH has vertices E(–8, 4), F(–4, 8), G(8, 4) and H(–4, –8). Graph the image of EFGH after a dilation centered at the origin with a scale factor of Multiply the x- and y-coordinates of each vertex by the scale factor, Example 3

Graph the preimage and image. Dilations in the Coordinate Plane Graph the preimage and image. Answer: E'(–2, 1), F'(–1, 2), G'(2, 1), H'(–1, –2) Example 3

Dilations in the Coordinate Plane B. Triangle ABC has vertices A(–2, -4), B(3, 2), C(0, 4). Graph the image of ABC after a dilation centered at (1, 2) with a scale factor of 1.5. Example 3

Graph the preimage and image. Answer: Dilations in the Coordinate Plane Graph the preimage and image. Answer: Example 3

Triangle ABC has vertices A(–1, 1), B(2, –2), and C(–1, –2) Triangle ABC has vertices A(–1, 1), B(2, –2), and C(–1, –2). Find the image of ΔABC after a dilation centered at the origin with a scale factor of 2. Sketch the preimage and the image. A. B. C. D. none of the above Example 3

Compositions of Dilations  

LESSON 9–6 Dilations