1. Midpoints: Segment Congruence
Chapter 1 Section 5
Midpoints: Segment Congruence

2. Warm-Up
1) Name two possible arrangements for G, H, and I on a segment if GH + GI = HI
2) Use the figure below to find each measure. D A C E
-10 -8 -6 -4 -2 2 4 6 8 10
a)  AC
b) DE
3) If M is between L and N, LN = 3x – 1, LM = 4, and MN = x – 1, find MN.   
4) What is the length of ST for S(-1, -1) and T(4, 6)?

3. 2) Use the figure below to find each measure.
1) Name two possible arrangements for G, H, and I on a segment if GH + GI = HI
 H, G, I or I, G, H
2) Use the figure below to find each measure. D A C E
-10 -8 -6 -4 -2 2 4 6 8 10
a)  AC
A= 1, C = 5
A – C
1 – 5 = -4
So AC is 4.
b) DE
D = -1, E = 8
D – E
-1 – 8 = -9
So DE is 9.

4. Use the segment addition Postulate. LM + MN = LN 4 + x -1 = 3x - 1
3) If M is between L and N, LN = 3x – 1, LM = 4, and MN = x – 1, find MN.   
Use the segment addition Postulate.
LM + MN = LN
4 + x -1 = 3x - 1
x + 3 = 3x - 1
3 = 2x - 1
4 = 2x
2 = x
Now plug 2 in for x in the equation for MN
MN = x - 1
MN = 2 - 1
MN = 1 L M N

5. 4) What is the length of ST for S(-1, -1) and T(4, 6)?
Distance Formula
d=√((x2 – x1)2 + (y2 – y1)2)
Pick one point to be x1 and y1 and the other point will be x2 and y2.
Let point S be x1 and y1 and point T be x2 and y2.
d=√((4 – -1)2 + (6 – -1)2)
d=√((4 + 1)2 + (6 + 1)2)
d=√((5)2 + (7)2)
d= √((25) + (49))
d= √(74)
So the distance between the two points is √(74) or about 8.6.

6. Vocabulary P M Q
Midpoint- The midpoint M of PQ is the point between P and Q such that PM = MQ
Segment bisector- Any segment, line, or plane that intersects a segment at its midpoint. Line L is a segment bisector.
Theorems- A statement that must be proven.
Proof- A logical argument in which each statement you make is backed up by a statement that is accepted as true. L

7. Vocabulary Cont.
Midpoint Formulas- On a number line, the coordinate of the midpoint of a segment whose endpoints have coordinates a and b is
(a + b)/2.
In a coordinate plane, the coordinates of the midpoint of a segment whose endpoints have the coordinates (x1, y1) and (x2, y2) are
[(x1 + x2)/2, (y1 + y2)/2].
Midpoint Theorem- If M is the midpoint of line AB, then Segment AM congruent to segment MB. A M B

8. Example 1: If the coordinate of H is -5 and the coordinate of J is 4, what is the coordinate of the midpoint of line HJ?
H and J are on a number line so use the equation
(a + b)/2.
Let point H be a and point J be b.
(a + b)/2
(-5 + 4)/2
-1/2
So the coordinate of the midpoint is at -1/2.

9. Example 2: If the coordinate of H is -10 and the coordinate of J is 2, what is the coordinate of the midpoint of line HJ?
H and J are on a number line so use the equation
(a + b)/2.
Let point H be a and point J be b.
(a + b)/2
( )/2
-8/2 -4
So the coordinate of the midpoint is at -4.

10. Example 3: Find the coordinates of the midpoint of line VW for V(3, -6) and W(7, 2).
V and W are on a coordinate plane so use the equation
[(x1 + x2)/2, (y1 + y2)/2].
Let point V be x1 and y1 and let point W be x2 and y2.
(x1 + x2)/2 = x-coordinate of the midpoint
(3 + 7)/2
10/2 5
(y1 + y2)/2 = y-coordinate of the midpoint
(-6 + 2)/2
(-4)/2 -2
So the midpoint of line VW is at the point (5,-2)

11. Example 4: Find the coordinates of the midpoint of line VW for V(4, -2) and W(8, 6).
V and W are on a coordinate plane so use the equation
[(x1 + x2)/2, (y1 + y2)/2].
Let point V be x1 and y1 and let point W be x2 and y2.
(x1 + x2)/2 = x-coordinate of the midpoint
(4 + 8)/2
12/2 6
(y1 + y2)/2 = y-coordinate of the midpoint
(-2 + 6)/2
(4)/2 2
So the midpoint of line VW is at the point (6,2)

12. Example 5: The midpoint of line RQ is P(4, -1)
Example 5: The midpoint of line RQ is P(4, -1). What are the coordinates of R if Q is at (3, -2)?
R and Q are on a coordinate plane so use the equation
[(x1 + x2)/2, (y1 + y2)/2].
Let point R be x1 and y1 and let point Q be x2 and y2.
(x1 + x2)/2 = x-coordinate of the midpoint
(x1 + 3)/2 = 4
x1 + 3 = 8
x1 = 5
(y1 + y2)/2 = y-coordinate of the midpoint
(y1 + -2)/2 = -1
(y1 + -2) = -2
y1 = 0
So point R is at (5,0).

13. Example 6: The midpoint of line RQ is P(4, -6)
Example 6: The midpoint of line RQ is P(4, -6). What are the coordinates of R if Q is at (8, -9)?
R and Q are on a coordinate plane so use the equation
[(x1 + x2)/2, (y1 + y2)/2].
Let point R be x1 and y1 and let point Q be x2 and y2.
(x1 + x2)/2 = x-coordinate of the midpoint
(x1 + 8)/2 = 4
x1 + 8 = 8
x1 = 0
(y1 + y2)/2 = y-coordinate of the midpoint
(y1 + -9)/2 = -6
(y1 + -9) = -12
y1 = -3
So point R is at (0,-3).

14. Plug 3 in for x in the equation for XY. XY = 16x – 6 XY = 16(3) – 6
Example 7: U is the midpoint of line XY. If XY = 16x – 6 and UY = 4x + 9, find the value of x and the measure of line XY. X U Y
Since U is the midpoint of line XY, we can use the midpoint formula. The midpoint formula tells us that XU is congruent or equal to UY. So XU + UY = XY; UY + UY = XY or 2(UY) = XY.
2( UY) = XY
2(4x + 9) = 16x – 6
8x + 18 = 16x – 6
18 = 8x – 6
24 = 8x
3 = x
Plug 3 in for x in the equation for XY.
XY = 16x – 6
XY = 16(3) – 6
XY = 48 – 6
XY = 42

15. Plug 4 in for x in the equation for XY. XY = 2x + 14 XY = 2(4) + 14
Example 8: U is the midpoint of line XY. If XY = 2x + 14 and UY = 4x - 5, find the value of x and the measure of line XY. X U Y
Since U is the midpoint of line XY, we can use the midpoint formula. The midpoint formula tells us that XU is congruent or equal to UY. So XU + UY = XY; UY + UY = XY or 2(UY) = XY.
2( UY) = XY
2(4x - 5) = 2x + 14
8x - 10 = 2x + 14
6x - 10 = 14
6x = 24
4 = x
Plug 4 in for x in the equation for XY.
XY = 2x + 14
XY = 2(4) + 14
XY =
XY = 22

16. Plug 8 in for x in either of the equations. XY = 2x + 11
Example 9: Y is the midpoint of line XZ. If XY = 2x + 11 and YZ = 4x - 5, find the value of x and the measure of line XZ. X Y Z
Since Y is the midpoint of line XZ, we can use the midpoint formula. The midpoint formula tells us that XY is congruent or equal to YZ. So XY + YZ = XZ; XY + XY = XZ or 2(XY) = XZ.
XY = YZ
2x + 11 = 4x - 5
11 = 2x - 5
16 = 2x
8 = x
Plug 8 in for x in either of the equations.
XY = 2x + 11
XY = 2(8) + 11
XY =
XY = 27
2(XY) = XZ
2(27) = XZ
54 = XZ

17. Plug 2 in for x in either of the equations. XY = -3x + 9
Example 9: Y is the midpoint of line XZ. If XY = -3x + 9 and YZ = 4x - 5, find the value of x and the measure of line XZ. X Y Z
Since Y is the midpoint of line XZ, we can use the midpoint formula. The midpoint formula tells us that XY is congruent or equal to YZ. So XY + YZ = XZ; XY + XY = XZ or 2(XY) = XZ.
XY = YZ
-3x + 9 = 4x - 5
9 = 7x - 5
14 = 7x
2 = x
Plug 2 in for x in either of the equations.
XY = -3x + 9
XY = -3(2) + 9
XY =
XY = 3
2(XY) = XZ
2(3) = XZ
6 = XZ