Geometry Section 1-3
- 1. Section 1-3 Locating Points and Midpoints
- 2. Essential Questions How do you find the midpoint of a segment? How do you locate a point on a segment given a fractional distance from one endpoint?
- 3. Vocabulary 1. Midpoint: The point on a segment that is halfway between the endpoints M = x1 + x2 2 , y1 + y2 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ for (x1 ,y1 ) and (x2 ,y2 ) 2. Segment Bisector: Any segment, line, or plane that intersects another segment at its midpoint
- 4. Example 1 Find the midpoint of AB for points A(3, 2) and B(6, 8). M = x1 + x2 2 , y1 + y2 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = 3 + 6 2 , 2 + 8 2 ⎛ ⎝⎜ ⎞ ⎠⎟ = 9 2 , 10 2 ⎛ ⎝⎜ ⎞ ⎠⎟ = 9 2 ,5 ⎛ ⎝⎜ ⎞ ⎠⎟ or 4.5,5( )
- 5. Example 2 Find the coordinates of U if F(−2, 3) is the midpoint of UO and O has coordinates of (8, 6). M = x1 + x2 2 , y1 + y2 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ (−2,3) = x + 8 2 , y + 6 2 ⎛ ⎝⎜ ⎞ ⎠⎟ −2 = x + 8 2 i2(2)i ( ) −4 = x + 8 x = −12 3 = y + 6 2 i22i 6 = y + 6 y = 0 U(−12,0)
- 6. Example 3 Find PQ if Q is the midpoint of PR. 2x + 3 4x − 1 P Q R 2x + 3 = 4x − 1 4 = 2x x = 2 PQ = 2x + 3 PQ = 2(2) + 3 PQ = 4 + 3 PQ = 7 units
- 7. Example 4 Find P if NM that is 1/3 the distance from N to M for points N(−3, −3) and M(2, 3). Horizontal change 1 3 x2 − x1 1 3 2 − (−3) 1 3 2 + 3 1 3 5 5 3 units x y M N
- 8. Example 4 Find P if NM that is 1/3 the distance from N to M for points N(−3, −3) and M(2, 3). Vertical change 1 3 y2 − y1 1 3 3 − (−3) 1 3 3 + 3 1 3 6 2 units x y M N
- 9. Example 4 Find P if NM that is 1/3 the distance from N to M for points N(−3, −3) and M(2, 3). Vertical change 2 units Horizontal change 5 3 units P(−3 + 5 3 ,−3 + 2) P(− 4 3 ,−1) x y M N
- 10. Example 5 Find F on AB such that the ratio of AF to FB is 2:3 for points A(−4, 6) and B(1, 2). 3:2 means 2 parts of AF and 3 parts of FB for a total of five parts. To go from A to F, we use 2/5 of the total distance from A to B.
- 11. Example 5 Find F on AB such that the ratio of AF to FB is 2:3 for points A(−4, 6) and B(1, 2). x y B A Horizontal change 2 5 x2 − x1 2 5 1− (−4) 2 5 1+ 4 2 5 5 2 units
- 12. Example 5 Find F on AB such that the ratio of AF to FB is 2:3 for points A(−4, 6) and B(1, 2). x y B A Vertical change 2 5 y2 − y1 2 5 2 − 6 2 5 −4 2 5 (4) 8 5 units
- 13. Example 5 Find F on AB such that the ratio of AF to FB is 2:3 for points A(−4, 6) and B(1, 2). x y B A Horizontal change 2 units Vertical change 8 5 units F(−4 + 2,6 − 8 5 ) F(−2, 22 5 )