CHAPTER 1SECTION 5

Midpoints: Segment Congruence

WARM-UP1) 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.

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)?

0 108642-2-8 -6 -4-10

D A EC

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.

0 108642-2-8 -6 -4-10

D A EC

a)  ACA= 1, C = 5A – C1 – 5 = -4So AC is 4.

b) DED = -1, E = 8D – E-1 – 8 = -9So 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 = LN4 + x -1 = 3x - 1x + 3 = 3x - 13 = 2x - 14 = 2x2 = x

Now plug 2 in for x in the equation for MNMN = x - 1MN = 2 - 1MN = 1

ML N

4) What is the length of ST for S(-1, -1) and T(4, 6)?

Distance Formulad=√((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.

VOCABULARYMidpoint- 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.

MP Q

L

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. 

MA 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)/210/25

(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)/212/26

(y1 + y2)/2 = y-coordinate of the midpoint(-2 + 6)/2(4)/22

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 = 4x1 + 3 = 8x1 = 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 = 4x1 + 8 = 8x1 = 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. UX 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) = XY2(4x + 9) = 16x – 68x + 18 = 16x – 618 = 8x – 624 = 8x3 = x

Plug 3 in for x in the equation for XY.XY = 16x – 6XY = 16(3) – 6XY = 48 – 6XY = 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. UX 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) = XY2(4x - 5) = 2x + 148x - 10 = 2x + 146x - 10 = 146x = 244 = x

Plug 4 in for x in the equation for XY.XY = 2x + 14XY = 2(4) + 14XY = 8 + 14XY = 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. YX 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 = YZ2x + 11 = 4x - 511 = 2x - 516 = 2x8 = xPlug 8 in for x in either of the equations.XY = 2x + 11XY = 2(8) + 11XY = 16 + 11XY = 27

2(XY) = XZ2(27) = XZ54 = 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. YX 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 - 59 = 7x - 514 = 7x2 = xPlug 2 in for x in either of the equations.XY = -3x + 9XY = -3(2) + 9XY = -6 + 9XY = 3

2(XY) = XZ2(3) = XZ6 = XZ