1. 1.3 Notes: Distance and Midpoints
EQ: How can I use distance formula and to find the length of a segment?

2. Vocab!
Distance
The distance between two points is the length of the segment with those points as its endpoints.

3. Plot (2 ,4 ) and (8 ,4 ). Find the distance.
6 units

4. Distance Formula (On the number line)
Vocab!
Distance Formula (On the number line)
𝑥 2 − 𝑥 1

5. Example 1
Use the number line to find QR.
−3−−6 =3

6. You Try! 1. Find the length of the segment: a. 𝐶𝐷 b. 𝐶𝐸 c. 𝐵𝐶 d. 𝐴𝐸
Do on your own!

7. You Try! 1. Find the length of the segment: a. 𝐶𝐷 b. 𝐶𝐸 c. 𝐵𝐶 d. 𝐴𝐸 13 3 8

8. Vocab!
Distance Formula
𝑑= ( 𝑥 2 − 𝑥 1 ) 2 + ( 𝑦 2 − 𝑦 1 ) 2

9. Example 2 Find the distance between E(–4, 1) and F(3, –1).
( 𝑥 1 , 𝑦 1 )
( 𝑥 2 , 𝑦 2 )
𝑑= ( 𝑥 2 − 𝑥 1 ) 2 + ( 𝑦 2 − 𝑦 1 ) 2
𝑑= (3−−4) 2 + (−1−−4) 2
𝑑= (7) 2 + (3) 2
𝑑= 49+9
𝑑= 53

10. You Try!
Find the distance between the two points. 1. (13 , 2) and (7 , 10)
Do on your own!
Answer = 10

11. Vocab! The point halfway between the endpoints of the segment.
Midpoint
The point halfway between the endpoints of the segment.

12. Example 3
Place a dot on -9, -3 and 5 labeling A, B, C respectively. a. Find the midpoint between point A and B. b. Find the midpoint between A and C. 3 3 A B C 7 7 6 -2

13. Midpoint Formula (on the number line)
Vocab!
Midpoint Formula (on the number line)
𝑥 1 + 𝑥 2 2

14. Example 4 𝑥 1 + 𝑥 2 2 1+ −4 2 =− 3 2 −2+ 4 2 =1 −4+2 2 =−1 −4+4 2 =0
Find the midpoint of the segment: a. 𝐶𝐴 b. 𝐵𝐸 c. 𝐴𝐷 d. 𝐴𝐸
𝑥 1 + 𝑥 2 2
1+ −4 2 =− 3 2
− =1
−4+2 2 =−1
−4+4 2 =0

15. Vocab!
Midpoint Formula  
𝑀( 𝑥 1 + 𝑥 2 2 , 𝑦 1 + 𝑦 2 2 )

16. Example 5
Find the coordinates of M, the midpoint of 𝐺𝐻 for G(8, –6), and H(–14, 12).
𝑀 8+ −14 2 , −6+12 2
𝑀(−3, 3)

17. You Try!
1: Find the coordinates of D if E(–6, 4) is the midpoint of 𝐷𝐹 and F has coordinates (–5, –3).
Do on your own!
Answer = (-7, 11)

18. Use equation, substitute what is given and solve
What if you were given a midpoint and a coordinate point at the end of a line segment…how would you find the other end of the line segment?
Use equation, substitute what is given and solve

19. Example 6
Find the coordinates of the missing endpoint if E is the midpoint of 𝐷𝐹 given that E(2,3) and F(5, 5)
2, 3 =( 5+ 𝑥 2 2 , 5+ 𝑦 2 2 ) F
2∙2= 5+ 𝑥 2 2 ∙2
4=5+ 𝑥 2
𝑥 2 =−1 E
2∙3= 5+ 𝑦 2 2 ∙2
6=5+ 𝑦 2
𝑥 2 =1

20. You Try!
1. Find the coordinates of the missing endpoint if E is the midpoint of 𝐷𝐹 . E( 1, 0) D(- 4, 3)
Do on your own!
Answer = (6, -3)

21. Vocab!
Segment Bisector
Any segment, line, or plane that intersects a segment at its midpoint

22. Example 8 Identify the segment bisector of 𝑃𝑄 . Then find PQ.
Segment bisector = MN
PQ = 1 7 8

23. Example 9
In the skate board design, 𝑉𝑊 bisects 𝑋𝑌 at point T and XT = 39.9 cm. Find XY.
XY = 39.9