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14 – 5 Dilations.

14 – 5 Dilations. Center Scale factor. A dilation D O, k maps any point P to a point P`, determined as follows: 1) If k > 0, P` lies on OP and OP` = |k|OP 2) If k<0, P` lies on the ray opposite OP and OP` = |K|OP 3) The center is its own image. P. O.

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14 – 5 Dilations.

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  1. 14 – 5 Dilations. Center Scale factor A dilation DO, k maps any point P to a point P`, determined as follows: 1) If k > 0, P` lies on OP and OP` = |k|OP 2) If k<0, P` lies on the ray opposite OP and OP` = |K|OP 3) The center is its own image P O |k| > 1 is an EXPANSION, expands the picture |k| < 1 is a CONTRACTION, shrinks the picture

  2. A dilation DO, k maps any point P to a point P`, determined as follows: 1) If k > 0, P` lies on OP and OP` = |k|OP 2) If k<0, P` lies on the ray opposite OP and OP` = |K|OP 3) The center is its own image B` B A` C` A C O 1) Draw a line through Center and vertex. 2) Extend or shrink segment by scale factor. (Technically by construction and common sense) 3) Repeat, then connect. |k| > 1 is an EXPANSION, expands the picture |k| < 1 is a CONTRACTION, shrinks the picture

  3. A dilation DO, k maps any point P to a point P`, determined as follows: 1) If k > 0, P` lies on OP and OP` = |k|OP 2) If k<0, P` lies on the ray opposite OP and OP` = |K|OP 3) The center is its own image B A B` C A` C` |k| > 1 is an EXPANSION, expands the picture |k| < 1 is a CONTRACTION, shrinks the picture O

  4. A dilation DO, k maps any point P to a point P`, determined as follows: 1) If k > 0, P` lies on OP and OP` = |k|OP 2) If k<0, P` lies on the ray opposite OP and OP` = |K|OP 3) The center is its own image B A C C` O A` |k| > 1 is an EXPANSION, expands the picture |k| < 1 is a CONTRACTION, shrinks the picture B` If you notice, a negative k in a way rotates the picture

  5. Theorem 14 – 5: A dilation maps any triangle to a similar triangle. Cor 1: A dilation maps an angle to a congruent angle. Cor 2: A dilation Do,k maps any segment to a parallel segment |k| times as long Cor 3: A dilation Do,k maps any polygon whose area is k2 times as large. Let’s discuss some justification here, types of invariancy.

  6. B B O A A C O C

  7. Remember, different slides for different periods. • F - HW #70: Pg 590: 31, 32, 34, 35, 36. Pg 596: 6—14 even, 15—19, 22, 23, 25, 26

  8. 0 period, 3 points curve, 2 for Ch 8, 1.5 for Ch 9. 6.5 total.

  9. 4th period, curve, 5. 2 pts for a mistake, 2 points for curve.

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