USING EQUATIONS FINDING THE LENGTH OF THE LINE SEGMENT

In the diagram given below, M is the midpoint of the segment. Find the indicated length.

Problem 1 :

Find length of AM.

Solution :

In the given diagram,

M is the midpoint of AC.

AM = x + 5, MC = 2x

x + 5 = 2x

 5 = 2x – x

5 = x

x = 5 substitute in AM = x + 5

AM = 5 + 5

AM = 10

So, the value of AM is 10.

Problem 2 :

Find the length of EM

Solution :

In the given diagram,

M is the midpoint of EG.

EM = 7x, MG = 8x - 6

7x = 8x - 6

 7x – 8x = – 6

-x = -6

x = 6

x = 6 Substitute in EM = 7x

EM = 7(6)

EM = 42

So, the length of EM is 42.

Problem 3 :

Solution :

In the given diagram,

M is the midpoint of JM.

JM = 6x + 7, ML = 4x + 5

6x + 7 = 4x + 5

 6x - 4x = 5 - 7

2x = -2

x = -1

x = -1 Substitute in JM = 6x + 7

JM = 6(-1) + 7

JM = -6 + 7

JM = 1

So, the value of JM is 1.

Problem 4 :

Find the length of PR.

Solution :

In the given diagram,

M is the midpoint of PR.

PM = 6x - 11, MR = 10x – 51

PM = MR

6x - 11 = 10x - 51

 6x – 10x = – 51 + 11

-4x = -40

x = 10

MR = 10x – 51

= 10(10) – 51

= 100 – 51

MR = 49

PM = 10

PR = 49 + 10

PR = 59

So, the length of PR is 59.

Problem 5 :

Find SU.

Solution :

In the given diagram,

M is the midpoint of SU.

SM = x + 15, MU = 4x – 45

SM = MU

x + 15 = 4x - 45

 x – 4x = – 45 - 15

-3x = -60

x = 60/3

x = 20

MU = 4x – 45

= 4(20) – 45

= 80 – 45

MU = 35

SM = 10

SU = 35 + 10

SU = 45

So, the value of SU is 45.

Problem 6 :

Find XZ.

Solution :

In the given diagram,

M is the midpoint of XZ.

XM = 2x + 35, MZ = 5x – 22

XM = MZ

2x + 35 = 5x - 22

 2x – 5x = – 22 - 35

-3x = -57

x = 57/3

x = 19

MZ = 5x – 22

= 5(19) – 22

= 95 – 22

MZ = 73

XM = 19

XZ = 73 + 19

XZ = 92

So, the length of XZ is 92.

Problem 7 :

Point M is the midpoint of VW. Find the length of VM.

Solution :

In the given diagram,

M is the midpoint of VW.

VM = 4x - 1, MW = 3x + 3

4x - 1 = 3x + 3

 4x – 3x = 3 + 1

x = 4

x = 4 substitute in VM = 4x - 1

VM = 4(4) - 1

VM = 15

So, the length of VM is 15.