Segment Addition | Math Coach Amy

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Key Ideas Find Total Length Find Segment Mixed With Quadratics Three Segments

AB + BC = AC

What to Remember

Segment Addition Postulate (numbers only)

If B is between A and C, then AB + BC = AC.

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AB + BC = AC

8 + 5 = AC

AC = 13

Segment Addition Postulate (variable side lengths)

Label the pieces and the whole. Solve for x, then substitute to get a length.

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AB = 3x + 1, BC = 2x, and AC = 21.

AB + BC = AC

(3x + 1) + 2x = 21

5x + 1 = 21

5x = 20

x = 4

AB = 3(4) + 1

AB = 13

Easy

Find Total Length

Add the two pieces. The first three diagrams are given. Draw the last three yourself.

Example

Point B is between A and C. AB = 9 and BC = 6. Find AC.

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AB + BC = AC

9 + 6 = AC

AC = 15

Answer: 15

Point Q is between P and R. PQ = 4 and QR = 11. Find PR.

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not drawn to scale

PQ + QR = PR

4 + 11 = PR

PR = 15

Answer: 15

Point K is between J and L. JK = 8 and KL = 8. Find JL.

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not drawn to scale

JK + KL = JL

JK = KL = 8

8 + 8 = JL

JL = 16

Answer: 16

Draw the segment. Point N is between M and P. MN = 7 and NP = 13. Find MP.

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MN + NP = MP

7 + 13 = MP

MP = 20

Answer: 20

Draw the segment. Point S is between R and T. RS = 15 and ST = 6. Find RT.

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RS + ST = RT

15 + 6 = RT

RT = 21

Answer: 21

Draw the segment. Point E is between D and F. DE = 10 and EF = 10. Find DF.

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DE + EF = DF

DE = EF = 10

10 + 10 = DF

DF = 20

Answer: 20

Easy

Find Segment

One piece is missing. The first three diagrams are given. Draw the last three yourself.

Example

Point Z is between X and Y. XY = 20 and XZ = 7. Find ZY.

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XZ + ZY = XY

7 + ZY = 20

ZY = 13

Answer: 13

Point B is between A and C. AC = 18 and BC = 5. Find AB.

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not drawn to scale

AB + BC = AC

AB + 5 = 18

AB = 13

Answer: 13

Point Q is between P and R. PR = 24 and PQ = 9. Find QR.

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not drawn to scale

PQ + QR = PR

9 + QR = 24

QR = 15

Answer: 15

Draw the segment. Point E is between D and F. DF = 30 and DE = 11. Find EF.

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DE + EF = DF

11 + EF = 30

EF = 19

Answer: 19

Draw the segment. Point K is between J and L. JL = 16 and KL = 7. Find JK.

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JK + KL = JL

JK + 7 = 16

JK = 9

Answer: 9

Draw the segment. Point S is between R and T. RT = 27 and RS = 14. Find ST.

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RS + ST = RT

14 + ST = 27

ST = 13

Answer: 13

Medium

Mixed

Draw the segment. Solve for x, then find the length.

Example

Draw the segment. Point B is between A and C. AB = 2x + 5, BC = x − 1, and AC = 19. Find AB.

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AB + BC = AC

(2x + 5) + (x − 1) = 19

3x + 4 = 19

3x = 15

x = 5

AB = 2(5) + 5

AB = 15

Answer: 15

Draw the segment. Point S is between R and T. RS = 4x − 3, ST = 2x + 1, and RT = 28. Find RS.

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RS + ST = RT

(4x − 3) + (2x + 1) = 28

6x − 2 = 28

6x = 30

x = 5

RS = 4(5) − 3

RS = 17

Answer: 17

Draw the segment. Point M is between L and N. LM = 3x + 2, MN = x + 6, and LN = 5x. Find LN.

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LM + MN = LN

(3x + 2) + (x + 6) = 5x

4x + 8 = 5x

8 = x

LN = 5(8)

LN = 40

Answer: 40

Draw the segment. Point B is between A and C. AB = x + 7, BC = 2x, and AC = 4x − 1. Find AC.

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AB + BC = AC

(x + 7) + 2x = 4x − 1

3x + 7 = 4x − 1

7 + 1 = 4x − 3x

8 = x

AC = 4(8) − 1

AC = 31

Answer: 31

Draw the segment. Point X is between W and Y. WX = 5x − 4, XY = x + 6, and WY = 4x + 8. Find WY.

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WX + XY = WY

(5x − 4) + (x + 6) = 4x + 8

6x + 2 = 4x + 8

2x = 6

x = 3

WY = 4(3) + 8

WY = 20

Answer: 20

Draw the segment. Point B is between A and C. AB = 4x − 1, BC = 2x + 7, and AC = 5x + 10. Find AC.

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AB + BC = AC

(4x − 1) + (2x + 7) = 5x + 10

6x + 6 = 5x + 10

x = 4

AC = 5(4) + 10

AC = 30

Answer: 30

Hard

With Quadratics

Draw the segment. Set the equation equal to 0, then factor.

Example

Draw the segment. Point B is between A and C. AB = x², BC = 5x, and AC = 36. Find AB.

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AB + BC = AC

x² + 5x = 36

Set the equation equal to 0.

x² + 5x − 36 = 0

For x² + bx + c = 0, find two numbers that multiply to c and add to b. Here c = −36 and b = 5. Those numbers are 9 and −4.

(x + 9)(x − 4) = 0

x = −9 or x = 4

x = −9 makes BC = 5(−9) = −45. A length cannot be negative, so reject x = −9.

x = 4

AB = 4²

AB = 16

Answer: 16

Draw the segment. Point Q is between P and R. PQ = x², QR = 3x, and PR = 40. Find PQ.

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PQ + QR = PR

x² + 3x = 40

Set the equation equal to 0.

x² + 3x − 40 = 0

For x² + bx + c = 0, find two numbers that multiply to c and add to b. Here c = −40 and b = 3. Those numbers are 8 and −5.

(x + 8)(x − 5) = 0

x = −8 or x = 5

x = −8 makes QR = 3(−8) = −24. A length cannot be negative, so reject x = −8.

x = 5

PQ = 5²

PQ = 25

Answer: 25

Draw the segment. Point B is between A and C. AB = x², BC = 2x, and AC = 48. Find AB.

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AB + BC = AC

x² + 2x = 48

Set the equation equal to 0.

x² + 2x − 48 = 0

For x² + bx + c = 0, find two numbers that multiply to c and add to b. Here c = −48 and b = 2. Those numbers are 8 and −6.

(x + 8)(x − 6) = 0

x = −8 or x = 6

x = −8 makes BC = 2(−8) = −16. A length cannot be negative, so reject x = −8.

x = 6

AB = 6²

AB = 36

Answer: 36

Draw the segment. Point S is between R and T. RS = x², ST = 4x, and RT = 45. Find RS.

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RS + ST = RT

x² + 4x = 45

Set the equation equal to 0.

x² + 4x − 45 = 0

For x² + bx + c = 0, find two numbers that multiply to c and add to b. Here c = −45 and b = 4. Those numbers are 9 and −5.

(x + 9)(x − 5) = 0

x = −9 or x = 5

x = −9 makes ST = 4(−9) = −36. A length cannot be negative, so reject x = −9.

x = 5

RS = 5²

RS = 25

Answer: 25

Draw the segment. Point E is between D and F. DE = x², EF = 4x, and DF = 60. Find DE.

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DE + EF = DF

x² + 4x = 60

Set the equation equal to 0.

x² + 4x − 60 = 0

For x² + bx + c = 0, find two numbers that multiply to c and add to b. Here c = −60 and b = 4. Those numbers are 10 and −6.

(x + 10)(x − 6) = 0

x = −10 or x = 6

x = −10 makes EF = 4(−10) = −40. A length cannot be negative, so reject x = −10.

x = 6

DE = 6²

DE = 36

Answer: 36

Draw the segment. Point X is between W and Y. WX = x², XY = x, and WY = 56. Find WX.

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WX + XY = WY

x² + x = 56

Set the equation equal to 0.

x² + x − 56 = 0

For x² + bx + c = 0, find two numbers that multiply to c and add to b. Here c = −56 and b = 1. Those numbers are 8 and −7.

(x + 8)(x − 7) = 0

x = −8 or x = 7

x = −8 makes XY = −8. A length cannot be negative, so reject x = −8.

x = 7

WX = 7²

WX = 49

Answer: 49

Hard

Three Segments

Draw the segment. Three pieces lie in a row.

Example

Draw the segment. Points A, B, C, and D are collinear in that order. AB = 6, BC = 9, and CD = 5. Find AD and AC.

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AC = AB + BC

AC = 6 + 9

AC = 15

AD = AB + BC + CD

AD = 6 + 9 + 5

AD = 20

Answer: AD = 20, AC = 15

Draw the segment. Points P, Q, R, and S are collinear in that order. PQ = 4, QR = 11, and RS = 8. Find PS and PR.

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PR = 4 + 11

PR = 15

PS = 4 + 11 + 8

PS = 23

Answer: PS = 23, PR = 15

Draw the segment. Points W, X, Y, and Z are collinear in that order. WX = 12, XY = 7, and YZ = 9. Find WZ and WY.

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WY = 12 + 7

WY = 19

WZ = 12 + 7 + 9

WZ = 28

Answer: WZ = 28, WY = 19

Draw the segment. Points A, B, C, and D are collinear in that order. AB = 2x + 1, BC = x + 4, CD = 5, and AD = 4x. Find AD.

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AB + BC + CD = AD

(2x + 1) + (x + 4) + 5 = 4x

3x + 10 = 4x

10 = x

AD = 4(10)

AD = 40

Answer: 40

Draw the segment. Points P, Q, R, and S are collinear in that order. PQ = 3x − 2, QR = x + 5, RS = 6, and PS = 5x + 1. Find PS.

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PQ + QR + RS = PS

(3x − 2) + (x + 5) + 6 = 5x + 1

4x + 9 = 5x + 1

8 = x

PS = 5(8) + 1

PS = 41

Answer: 41

Draw the segment. Points A, B, C, and D are collinear in that order. AB = x + 3, BC = 2x + 1, CD = x − 2, and AD = 5x − 6. Find AD.

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AB + BC + CD = AD

(x + 3) + (2x + 1) + (x − 2) = 5x − 6

4x + 2 = 5x − 6

2 + 6 = 5x − 4x

8 = x

AD = 5(8) − 6

AD = 34

Answer: 34

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