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Chapter 1 Section 5

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Chapter 1Section 5

Midpoints: Segment Congruence

- 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

0

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)?

- 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

0

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.

- 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

- 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=√((x2 – x1)2 + (y2 – y1)2)
- 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.

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

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

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.

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

(-10 + 2)/2

-8/2

-4

So the coordinate of the midpoint is at -4.

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)

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)

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).

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).

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

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 = 8 + 14

XY = 22

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 = 16 + 11

XY = 27

2(XY) = XZ

2(27) = XZ

54 = XZ

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 = -6 + 9

XY = 3

2(XY) = XZ

2(3) = XZ

6 = XZ