Geometry · Unit 2
A midpoint cuts a segment into two congruent pieces.
Jump to section:
midpoint: two congruent pieces
AM = MB
Segment Bisector (numbers only)
A midpoint splits a segment into two congruent pieces.
If M is the midpoint of AB and AM = 9, then MB = 9 and AB = 18.
not drawn to scale
AM = MB
AM = MB = 9
AB = 9 + 9
AB = 18
Equidistant
If a point is equidistant from A and B, it lies on the perpendicular bisector of AB. It is the midpoint only when it is also on the segment.
Not drawn to scale.
M is on the segment, and MA = MB, so M is the midpoint. P is off the segment. PA can equal PB without P being the midpoint.
Segment Bisector (variable side lengths)
The two pieces are equal. Set them equal, solve, then plug back.
not drawn to scale
AM = 3x + 2 and MB = x + 10.
AM = MB
3x + 2 = x + 10
2x = 8
x = 4
AM = 3(4) + 2
AM = 14
AB = 14 + 14
AB = 28
Double the piece you know. The first three diagrams are given. Draw the last three yourself.
not drawn to scale
not drawn to scale
AM = MB
AM = MB = 9
AM + MB = AB
AB = 9 + 9
AB = 18
Answer: 18
not drawn to scale
not drawn to scale
CM = MD
CM = MD = 7
CM + MD = CD
CD = 7 + 7
CD = 14
Answer: 14
not drawn to scale
not drawn to scale
PM = MQ
PM = MQ = 15
PM + MQ = PQ
PQ = 15 + 15
PQ = 30
Answer: 30
not drawn to scale
RM = MS
RM = MS = 6
RM + MS = RS
RS = 6 + 6
RS = 12
Answer: 12
not drawn to scale
JM = MK
JM = MK = 11
JM + MK = JK
JK = 11 + 11
JK = 22
Answer: 22
not drawn to scale
WM = MX
WM = MX = 4
WM + MX = WX
WX = 4 + 4
WX = 8
Answer: 8
The whole segment is given. Find one congruent piece. Draw the last three yourself.
not drawn to scale
not drawn to scale
PM = MQ
PM = 242
PM = 12
Answer: 12
not drawn to scale
not drawn to scale
AM = MB
AM = 182
AM = 9
Answer: 9
not drawn to scale
not drawn to scale
EM = MF
EM = 302
EM = 15
Answer: 15
not drawn to scale
JM = MK
JM = 162
JM = 8
Answer: 8
not drawn to scale
RM = MS
RM = 422
RM = 21
Answer: 21
not drawn to scale
CM = MD
CM = 102
CM = 5
Answer: 5
Draw the segment. Set the pieces equal, solve, then find the length.
not drawn to scale
AM = MB
2x + 5 = 4x − 7
5 + 7 = 4x − 2x
12 = 2x
x = 6
AM = 2(6) + 5
AM = 17
AB = 17 + 17
AB = 34
Answer: 34
not drawn to scale
AM = MB
5x − 3 = 3x + 7
2x = 10
x = 5
AM = 5(5) − 3
AM = 22
AB = 22 + 22
AB = 44
Answer: 44
not drawn to scale
AM = MB
x + 12 = 3x
12 = 2x
x = 6
AM = 18
AB = 18 + 18
AB = 36
Answer: 36
not drawn to scale
AM = MB
4x + 1 = 2x + 11
2x = 10
x = 5
AM = 4(5) + 1
AM = 21
AB = 42
Answer: 42
not drawn to scale
AD is half of AC, so 2(AD) = AC.
2(x + 8) = 4x + 2
2x + 16 = 4x + 2
14 = 2x
x = 7
AC = 4(7) + 2
AC = 30
Answer: 30
not drawn to scale
2(AM) = AB
2(5x − 4) = 6x + 10
10x − 8 = 6x + 10
4x = 18
x = 92
AM = 592 − 4
AM = 452 − 82
AM = 372
Answer: 372
Draw the segment. Set the equation equal to 0, then factor.
not drawn to scale
AM = MB
x² − 5 = 4x − 8
Set the equation equal to 0.
x² − 4x + 3 = 0
For x² + bx + c = 0, find two numbers that multiply to c and add to b. Here c = 3 and b = −4. Those numbers are −1 and −3.
(x − 1)(x − 3) = 0
x = 1 or x = 3
x = 1 makes AM = 1 − 5 = −4. A length cannot be negative, so reject x = 1.
x = 3
AM = 3² − 5 = 4
AB = 4 + 4 = 8
Answer: 8
not drawn to scale
AM = MB
x² − 10 = 3x
Set the equation equal to 0.
x² − 3x − 10 = 0
For x² + bx + c = 0, find two numbers that multiply to c and add to b. Here c = −10 and b = −3. Those numbers are −5 and 2.
(x − 5)(x + 2) = 0
x = 5 or x = −2
x = −2 makes AM = 4 − 10 = −6. A length cannot be negative, so reject x = −2.
x = 5
AM = 25 − 10 = 15
AB = 30
Answer: 30
not drawn to scale
AM = MB
x² − 8 = 2x
Set the equation equal to 0.
x² − 2x − 8 = 0
For x² + bx + c = 0, find two numbers that multiply to c and add to b. Here c = −8 and b = −2. Those numbers are −4 and 2.
(x − 4)(x + 2) = 0
x = 4 or x = −2
x = −2 makes AM = 4 − 8 = −4. A length cannot be negative, so reject x = −2.
x = 4
AM = 16 − 8 = 8
AB = 16
Answer: 16
not drawn to scale
AM = MB
x² − 12 = 4x
Set the equation equal to 0.
x² − 4x − 12 = 0
For x² + bx + c = 0, find two numbers that multiply to c and add to b. Here c = −12 and b = −4. Those numbers are −6 and 2.
(x − 6)(x + 2) = 0
x = 6 or x = −2
x = −2 makes AM = 4 − 12 = −8. A length cannot be negative, so reject x = −2.
x = 6
AM = 36 − 12 = 24
AB = 48
Answer: 48
not drawn to scale
AM = MB
x² − 18 = 3x
Set the equation equal to 0.
x² − 3x − 18 = 0
For x² + bx + c = 0, find two numbers that multiply to c and add to b. Here c = −18 and b = −3. Those numbers are −6 and 3.
(x − 6)(x + 3) = 0
x = 6 or x = −3
x = −3 makes AM = 9 − 18 = −9. A length cannot be negative, so reject x = −3.
x = 6
AM = 36 − 18 = 18
AB = 36
Answer: 36
not drawn to scale
AM = MB
x² − 21 = 4x
Set the equation equal to 0.
x² − 4x − 21 = 0
For x² + bx + c = 0, find two numbers that multiply to c and add to b. Here c = −21 and b = −4. Those numbers are −7 and 3.
(x − 7)(x + 3) = 0
x = 7 or x = −3
x = −3 makes AM = 9 − 21 = −12. A length cannot be negative, so reject x = −3.
x = 7
AM = 49 − 21 = 28
AB = 56
Answer: 56
Draw the segment. Midpoints can make a chain of equal pieces.
not drawn to scale
D is the midpoint of AC, so AD = DC.
C is the midpoint of BD, so DC = CB.
AD = DC = CB.
BD is two of those pieces.
Each piece is 202 = 10.
AB is three pieces.
AB = 10 + 10 + 10
AB = 30
Answer: 30
not drawn to scale
AD = DC and DC = CB, so the three pieces are equal.
BD is two pieces, so each piece is 7.
AB = 7 + 7 + 7
AB = 21
Answer: 21
not drawn to scale
N is the midpoint of AM, so AN = NM.
M is the midpoint of AB, so AM = MB.
MB is two small pieces, and NB is three small pieces.
3 small pieces = 18, so each small piece is 6.
AB is four small pieces.
AB = 24
Answer: 24
not drawn to scale
Let DE = ER = x. Then DR = 2x and PD = 2x.
PE = PD + DE = 3x.
3x = 21
x = 7
PR = 4x = 28
Answer: 28
not drawn to scale
AB = BD = 362 = 18.
BC = CD = 182 = 9.
AC = AB + BC
AC = 18 + 9
AC = 27
Answer: 27
not drawn to scale
Let MN = NY = x. Then MY = 2x and XM = 2x.
XN = XM + MN = 3x.
3x = 30
x = 10
XY = 4x = 40
Answer: 40
A point the same distance from both endpoints is on the perpendicular bisector.
Equidistant from the endpoints means the point is on the perpendicular bisector.
The midpoint is the point where that line meets the segment.
A point off the segment can be equidistant and still not be the midpoint.
Answer: false
On the segment, there is only one point equidistant from the endpoints.
That point is the midpoint.
Answer: true
That is the perpendicular bisector theorem.
Each of those points is the same distance from A and from B.
Answer: true
PA = PB puts P on the perpendicular bisector.
The midpoint has to be on AB. P is not.
Answer: false
The midpoint cuts AB into two congruent pieces.
Those pieces are the distances to A and to B.
Answer: true
That is the converse of the perpendicular bisector theorem.
P may be the midpoint, or it may be another point on that line.
Answer: true
Need help with a midpoint?
Book a Session with Amy