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

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

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

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

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

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

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

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

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

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

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

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

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

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 −2+ 4 2 =1 −4+2 2 =−1 −4+4 2 =0

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

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)

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)

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

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

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)

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

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

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