1. 1.3 Key Concepts

2. The Point that divides the segment into two congruent segments.
Midpoint
The Point that divides the segment into two congruent segments.

3. Segment Bisector
A Point, Ray, Line, Line Segment, or Plane that intersects the segment at its Midpoint

4. Midpoint Formula
[(x1 + x2)/2], [(y1 + y2)/2]

5. Distance Formula
AB = √[(x2-x1)2 + (y2-y1)2]

6. Point T is the midpoint of XY . So, XT = TY = 39.9 cm.
EXAMPLE 1
Find segment lengths
In the skateboard design, VW bisects XY at point T, and XT = 39.9 cm. Find XY.
Skateboard
SOLUTION
Point T is the midpoint of XY . So, XT = TY = 39.9 cm.
XY = XT + TY
Segment Addition Postulate =
Substitute.
= 79.8 cm
Add.

7. Use algebra with segment lengths
EXAMPLE 2
Use algebra with segment lengths
Point M is the midpoint of VW . Find the length of VM .
ALGEBRA
SOLUTION
STEP 1
Write and solve an equation. Use the fact that VM = MW.
VM = MW
Write equation.
4x – 1 = 3x + 3
Substitute.
x – 1 = 3
Subtract 3x from each side.
x = 4
Add 1 to each side.

8. EXAMPLE 2
Use algebra with segment lengths
STEP 2
Evaluate the expression for VM when x = 4.
VM = 4x – 1 = 4(4) – 1 = 15
So, the length of VM is 15.
Check: Because VM = MW, the length of MW should be 15. If you evaluate the expression for MW, you should find that MW = 15.
MW = 3x + 3 = 3(4) +3 = 15

9. GUIDED PRACTICE
for Examples 1 and 2
In Exercises 1 and 2, identify the segment bisector of PQ . Then find PQ. 1. 3 4
ANSWER
MN;

10. GUIDED PRACTICE
for Examples 1 and 2
In Exercises 1 and 2, identify the segment bisector of PQ . Then find PQ. 2.
line l ; 11 5 7
ANSWER

11. EXAMPLE 3
Use the Midpoint Formula
a. FIND MIDPOINT The endpoints of RS are R(1,–3) and S(4, 2). Find the coordinates of the midpoint M.

12. EXAMPLE 3
Use the Midpoint Formula
SOLUTION
a. FIND MIDPOINT Use the Midpoint Formula. 2 5
1 + 4
– 3 + 2 = , M – 1
The coordinates of the midpoint M are 1 , – 5 2
ANSWER

13. EXAMPLE 3
Use the Midpoint Formula
b. FIND ENDPOINT The midpoint of JK is M(2, 1). One endpoint is J(1, 4). Find the coordinates of endpoint K.

14. EXAMPLE 3
Use the Midpoint Formula
SOLUTION
FIND ENDPOINT Let (x, y) be the coordinates of endpoint K. Use the Midpoint Formula.
STEP 1
Find x.
STEP 2
Find y.
1+ x 2 =
4+ y 1 2 =
1 + x = 4
4 + y = 2
x = 3
y = – 2
The coordinates of endpoint K are (3, – 2).
ANSWER

15. GUIDED PRACTICE
for Example 3
3. The endpoints of AB are A(1, 2) and B(7, 8). Find the coordinates of the midpoint M.
ANSWER
(4,5)
4. The midpoint of VW is M(– 1, – 2). One endpoint is W(4, 4). Find the coordinates of endpoint V.
ANSWER
(– 6, – 8)

16. EXAMPLE 4
Standardized Test Practice
SOLUTION
Use the Distance Formula. You may find it helpful to draw a diagram.

17. Standardized Test Practice
EXAMPLE 4
Standardized Test Practice
(x – x ) + (y – y ) 2 1 RS =
Distance Formula
[(4 – 2)] + [(–1) –3] 2 =
Substitute.
(2) + (–4 ) 2 =
Subtract.
4+16 =
Evaluate powers. 20 =
Add.
4.47 ~ =
Use a calculator to approximate the square root.
The correct answer is C.
ANSWER

18. GUIDED PRACTICE
for Example 4
6. What is the approximate length of AB , with endpoints A(–3, 2) and B(1, –4)?
6.1 units
7.2 units
8.5 units
10.0 units B
ANSWER