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Similarity

Similarity. Chapter 6. 6.1 – Use Similar Polygons. Similar Polygons Two polygons are similar if corresponding angles are congruent and corresponding sides length are proportional Notation used is a tilde ( ~ ) Example 1: ΔRST ~ Δ XYZ RS=20, ST=30, TR=25 XY=12, YZ=18, ZX=15

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Similarity

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  1. Similarity Chapter 6

  2. 6.1 – Use Similar Polygons • Similar Polygons • Two polygons are similar if corresponding angles are congruent and corresponding sides length are proportional • Notation used is a tilde ( ~ ) • Example 1: • ΔRST ~ ΔXYZ • RS=20, ST=30, TR=25 XY=12, YZ=18, ZX=15 • List all pairs of congruent angles • Check that the ratios of corresponding side lengths are equal • Write the ratios of the corresponding side lengths in a statement of proportionality

  3. Scale Factor • Statement of proportionality • Any pair of ratios form a true proportion • Scale factor • Ratio of the lengths of two corresponding sides of two similar polygons • Example 3 • ΔDEF ~ ΔMNP • Find the value of x • GP #2-3, page 359

  4. Perimeters • Ratio of lengths in similar polygons is the same as the scale factor, this is also true for the perimeters of the polygons • 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 • If

  5. Example 4 • A town is building a new swimming pool. An Olympic pool is rectangular with length 50 meters and width 25 meters. The new pool will be similar in shape, but only 40 meters long. • Find the scale factor of the new pool to an Olympic pool • Find the perimeter of an Olympic pool and the new pool • GP #4-6, page 360

  6. 6.3 – Prove Triangles Similar by AA • 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 • Example 1 • Example 2

  7. Indirect Measurement • A useful way to find measurements indirectly is by using similar triangles • Example 3 • A flagpole casts a shadow that is 50 feet long. At the same time, a woman standing nearby who is five feet four inches tall casts a shadow that is 40 inches long. • How tall is the flagpole to the nearest foot? GP #4-5, page 377

  8. 6.4 – Prove Triangles Similar by SSS and SAS • In addition to using congruent corresponding angles, to show that two triangles are similar, you can use proportional corresponding side lengths • Theorem 6.2 – Side-Side-Side (SSS) Similarity Theorem • If the corresponding side lengths of two triangles are proportional, then the triangles are similar • Example 1 • Example 2

  9. SAS Similarity Theorem • 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 • Example 4 (on board) • Tell what method you would use to show that the triangles are similar GP #1-2, page 383 & #3-4, page 385

  10. 6.5 – Use Proportionality Theorems • The Midsegment theorem is a special case of the Triangle Proportionality theorem and its converse • 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

  11. Theorem 6.5 • Theorem 6.5 – Converse of Triangle Proportionality Theorem • If a line divides two sides of a triangle proportionally, then it is parallel to the third side • Example 1 • GP #1-2, page 392

  12. Theorems • 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

  13. Examples • Example 3 • Example 4 (on board) • Use the given information to find the length of segment RS • GP #3-4, page 393

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