Chapter 6: Similarity

Transcription

1 Name: Chapter 6: Similarity Guided Notes Geometry Fall Semester

2 CH. 6 Guided Notes, page Ratios, Proportions, and the Geometric Mean Term Definition Example ratio of a to b equivalent ratios proportion means extremes Property of Proportions 1: Cross Products Property In a proportion, the product of the extremes equals the product of the means. The geometric mean of two positive numbers geometric mean a and b is the positive number x that satisfies a x =. So, x 2 = ab x b and x = ab.

3 Examples: CH. 6 Guided Notes, page 3 1. Simplify the ratio. a) 76cm : 8cm b) 4ft : 24in 2. You are painting barn doors. You know that the perimeter of the doors is 64ft. and that the ratio of the length to the height is 3 : 5. Find the area of the doors. Extended Ratios 3. The measures of the angles in "BCD are in the extended ratio 2 : 3 : 4. Find the measures of the angles.

4 4. Solve the proportion. a) 3 4 = x 16 b) 3 x +1 = 2 x CH. 6 Guided Notes, page 4 Geometric Mean 5. Find the geometric mean of 16 and 48.

5 CH. 6 Guided Notes, page Use Proportions to Solve Geometry Problems Term Definition Example Property of Proportions 2: Reciprocal Property If two ratios are equal, then their reciprocals are also equal. If you interchange the means of a Property of Proportions 3 proportion, then you form another true proportion. In a proportion, if you add the value of each Property of Proportions 4 ratio s denominator to its numerator, then you form another true proportion. scale drawing scale

6 Examples: CH. 6 Guided Notes, page 6 1. In the diagram, AC DF = BC. Write a proportion based on the side lengths of the EF triangles, and then write three other proportions based on the Properties of Proportions. Proportion: Reciprocal Property: Prop. Of Prop. 3: Prop. Of Prop. 4: 2. In the diagram, JL LH = JK KG. Find JH and JL.

7 CH. 6 Guided Notes, page 7 Finding the scale of a drawing. 3. The length of the key in the scale drawing is 7 cm. The length of the actual key is 4 cm. What is the scale of the drawing? (To find the scale, write the ratio of a length in the drawing to the actual length. Then rewrite the ratio so that the denominator is 1.) Using a scale drawing. 4. The scale of the map at right is 1 inch : 8 miles. Find the actual distance from Westbrook to Cooley.

8 6.3 Use Similar Polygons CH. 6 Guided Notes, page 8 Term Definition Example similar polygons 1. Corresponding Angles are congruent 2. Ratios of Corresponding Sides are equal statement of proportionality scale factor Theorem 6.1 Perimeters of Similar Polygons If two polygons are similar, then the ratio of their perimeters is equal to the ratios of their corresponding side lengths. Corresponding Lengths in Similar Polygons If two polygons are similar, then the ratio of any two corresponding lengths in the polygons is equal to the scale factor of the similar polygons. similarity and congruence

9 CH. 6 Guided Notes, page 9 Examples: 1. In the diagram, "ABC ~ "DEF. a) List all pairs of congruent angles. b) Check that the ratios of corresponding side lengths are equal. c) Write the ratios of the corresponding side lengths in a statement of proportionality. 2. Determine whether the polygons are similar. If they are, write a similarity statement and find the scale factor of ABCD to JKLM.

10 CH. 6 Guided Notes, page In the diagram, "BCD ~ "RST. Find the value of x. 4. A larger cement court is being poured for a basketball hoop in place of a smaller one. The court will be 20 feet wid and 25 feet long. The old court is similar in shape, but only 16 feet wide. a) Find the scale factor of the new court to the old court. b) Find the perimeters of the new court and the old court. 5. In the diagram, "FGH ~ "JGK. Find the length of the altitude GL.

11 6.4 Prove Triangles Similar by AA CH. 6 Guided Notes, page 11 Term Definition Example Postulate 22 Angle-Angle (AA) Similarity Postulate If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar. Examples: 1. Determine whether the triangles are similar. If they are, write a similarity statement. Explain your reasoning. 2. Show that the two triangles are similar. a). "RTV and "RQS b) "LMN and "NOP

12 CH. 6 Guided Notes, page Prove Triangles Similar by SSS and SAS Term Definition Example Theorem 6.2 Side-Side-Side (SSS) Similarity Theorem If the corresponding side lengths of two triangles are proportional, then the triangles are similar. Theorem 6.3 Side-Angle-Side (SAS) Similarity Theorem If an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar. Examples: 1. Is either "DEF or "GHJ similar to "ABC?

13 2. Find the value of x that makes "ABC ~ "DEF. CH. 6 Guided Notes, page You are drawing a design for a birdfeeder. Can you construct the top so it is similar to the bottom using the angle measure and lengths shown? Choose a method 4. Tell what method you would use to show that the triangles are similar.

14 6.6 Use Proportionality Theorems CH. 6 Guided Notes, page 14 Term Definition Example Theorem 6.4 Triangle Proportionality Theorem If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally. Theorem 6.5 Converse of the Triangle Proportionality Theorem If a line divides two sides of a triangle proportionally, then it is parallel to the third side. Theorem 6.6 If three parallel lines intersect two transversals, then they divide the transversals proportionally. Theorem 6.7 If a ray bisects an angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the other two sides.

15 Examples: CH. 6 Guided Notes, page In the diagram QS //UT, RQ = 10, RS = 12, and ST = 6. What is the length of QU? 2. A farmer s land is divided by a newly constructed interstate. The distances shown are in meters. Find the distance CA between the north border and the south border of the farmer s land. (Use Theorem 6.6) 3. In the diagram, "DEG # "GEF. Use the given side lengths to find the length of DG.

16 6.7 Perform Similarity Transformations CH. 6 Guided Notes, page 16 Term Definition Example dilation similarity transformation center of dilation scale factor of a dilation reduction enlargement Coordinate Notation for a Dilation