**Dilations**

**Why dilate?** • Some photographers prefer traditional cameras and film to produce negatives. From these negatives, photographers can create scaled drawings.

**Why dilate?** • Scientists need to understand dilations when examining organisms under a microscope.

**Why dilate?** • Crime labs investigate DNA, hair samples, and fingerprints to solve cases.

**Why dilate?** • Architects design buildings and structures on the computer and create blueprints from which builders and engineers construct.

**Key Concepts** • The image and pre-image of a figure are similar • A dilation with center C and scale factor k • This is a type of transformation, not rigid, but non-rigid • A dilation is a reduction if 0<k<1 • A dilation is an enlargement if k>1

**Similar Figures** (Not exactly the same, but pretty close!)

**Congruent Figures** In order to be congruent, two figures must be the exact same size and same shape (isometry)

**Similar Figures** Similar figures must be the same shape, but their sizes may be different. Their sides must be proportional.

**9 cm** W Z 3 cm A D 2 cm 2 cm 6 cm 6 cm B C 3 cm Y X 9 cm In the rectangles above, one proportion is = , or = . AB WX AD WZ 2 6 3 9 Similar Figures If you cannot use corresponding side lengths to write a proportion, or if corresponding angles are not congruent, then the figures are not similar.

**SIZES** Although the size of the two shapes are different, the sizes of the two shapes must differ by a factor. 4 2 3 3 6 6 1 2 In this case, the factor is 2.

**Enlargements** When you have a photograph enlarged, you make a similar photograph. X 3

**Reductions** A photograph can also be shrunk to produce a slide. 4

**Similarity is used to answer real life questions.** Suppose that you wanted to find the height of this tree. Unfortunately all that you have is a tape measure, and you are too short to reach the top of the tree.

**You can measure the length of the tree’s shadow.** 10 feet

**Then, measure the length of your shadow.** 10 feet 2 feet

**If you know how tall you are, then you can determine how** tall the tree is. 6 ft 10 feet 2 feet

**The tree must be 30 ft tall. Boy, that’s a tall tree!** 6 ft 10 feet 2 feet

**Determine the length of the unknown side. ** 15 12 ? 4 3 9

**Determine the length of the unknown side.** ? 2 24 4

**Sometimes the factor between 2 figures is not obvious and** some calculations are necessary. 15 12 8 10 18 12 ? =

**When changing the size of a figure, will the angles of the** figure also change? ? 40 70 ? ? 70

**Nope! Remember, the sum of all 3 angles in a triangle MUST** add to 180 degrees.If the size of theangles were increased,the sum would exceed180degrees. 40 40 70 70 70 70

**We can verify this fact by placing the smaller triangle** inside the larger triangle. 40 40 70 70 70 70

**The 40 degree angles are congruent.** 40 70 70 70 70

**The 70 degree angles are congruent.** 40 40 70 70 70 70 70

**The other 70 degree angles are congruent.** 4 40 70 70 70 70 70

**Find the length of the missing side.** 50 30 ? 6 40 8

**This looks messy. Let’s translate the two triangles.** 50 30 ? 6 40 8

**Now “things” are easier to see.** 50 30 ? 6 40 8

**The common factor between these triangles is 5.** 50 30 ? 6 40 So the length of the missing side is…? 8

**Similar Figures** • Corresponding sides have lengths that are proportional. • Corresponding angles are congruent.

**Dilations in the Coordinate Plane** This is the rule only when the dilation is centered at the origin. We will dilate figures that are not centered at the origin as well.

**Guided Practice** Triangle DEF has vertices D(-1,-2), E(-2,2), and F(1,1). Find the coordinates of triangle D’E’F’, after a dilation of -2.

**Real-World Application**

**Homework:** • Finish the back of the class work sheet • Tomorrow we will be incorporating the translations we learned from the last unit, so PLEASE REVIEW THEM