Transcript Warm-up
Warm-up 1. D is between B and E. ED = x, BD = 30 and EB = 4x + 6. Find ED and EB. 2. E bisects segment DF. DE = 2x, and EF = 8x – 3. Find DE, EF, and DF. Discovering the Midpoint Formula What do you think a midpoint is? Write down a quick definition and discuss it with your partner. Which point do you think is the midpoint in the picture to the right? How do you know for sure it’s the midpoint? What if the red marks were not there? Would you know for sure it was still a midpoint? Discovering the Midpoint Formula So, M is the midpoint. What if the points were on a number line? P was located at 2 and Q was located at 8. Where would M be located? P M Q How did you know where M was located? How can you use math to find out M’s location? Some people may have just counted to find M, But what if the numbers were so big, it was hard to count…. Discovering the Midpoint Formula If M was still a midpoint, how could we use math to find it’s location now? P -35 -1 M 33 Q 67 101 If you are still stuck, let’s relate it to your grades. Say you got a 90 on one paper and a 70 on another paper. What would your final grade be? How did you use your math to get your final grade? Could you use this same idea to find M? 135 Discovering the Midpoint Formula Everything we have dealt with was 1-dimensional; we were using a number line. What if we moved to 2-D? How would we find a midpoint on a coordinate plane without counting? If you are stuck, (4, 10) estimate where you think the midpoint is. y Did you get (-2, 1)? (-8, -8) (-2, 1) x How can you use math to get these coordinates? Discovering the Midpoint Formula Just AVERAGE!!! Average your x values: 8 4 2 2 y Then average your y values: (4, 10) (-8, -8) (-2, 1) x 8 10 1 2 ANSWER: (-2, 1) Midpoint Formula • (The average of the x’s and the average of the y’s) x1 x2 y1 y2 ( , ) 2 2 Examples: Find the coordinates of the midpoint of each segment: 1. (-8,5) and (3,-6) 2. (4,6) and (6,4) 3. (8,6) and (22,20) On your own. 4. (0,5) and (6,7) 5. (1,-4) and (9,3) 6. (6,0) and (3,5) Discovering the Midpoint Formula One endpoint of a segment is (-2, 8). The midpoint is (4, -1). What are the coordinates of the other endpoint? y (-2, 8) (4, -1) x Example: M is the midpoint of AB . A has the coordinates (2,2) and M has the coordinates (4,-3). Find the coordinates of B. Extra Examples: M is the midpoint of XY . X has coordinates (2,7) and M has coordinates (6,1). Find the coordinates of Y. S is the midpoint of RT. R has coordinates (-6,-1), and S has coordinates (-1,1). Find the coordinates of T.