Segment addition postulate

The segment addition postulate states the following for 3 points that are collinear.  

Consider the segment on the right.

If 3 points A, B, and C are collinear and B is between A and C, then

AB + BC = AC

Using the segment addition postulate to solve a problem.

Suppose AC =  48, find the value of x. Then, find the length of AB and the length of BC.

AB + BC = AC

( 2x - 4 ) + ( 3x + 2 ) = 48

2x + 3x - 4 + 2 = 48

5x - 4 + 2 = 48

Add 4 to both sides of the equation.

5x - 4 + 4 + 2 = 48 + 4

5x + 2 = 52

Subtract 2 from both sides.

5x + 2 - 2 = 52 - 2

5x = 50

Divide both sides by 5

5x / 5 = 50 / 5

x = 10

Now that we have the value of x, we can find the length of AB and the length of BC.

AB = 2x - 4

AB = 2 × 10 - 4

AB = 20 - 4 = 16

The length of AB is 16

BC = 3x + 2

BC = 3 × 10 + 2

BC = 30 + 2 = 32

The length of BC is 32.

Segment addition postulate and the midpoint

Suppose XA = 3x and AY =  4x - 6. If A is the midpoint of XY, what is the length of XY?

           3x                       4x - 6
_________________________________
X                         A                         Y

The trick in this problem is to see that if A is the midpoint, then XA = AY.

Since XA = AY, 3x = 4x - 6

Subtract 3x from both sides.

3x - 3x = 4x - 3x - 6

0  = x - 6

Add 6 to both sides of the equation.

0 + 6 = x - 6 + 6

6 = x

To compute XA, you can either use 3x or 4x - 6

Using 3x, we get XA = AY = 3 × 6 = 16

Using 4x - 6, we get XA = AY = 3 × 6 - 6 = 18 - 6 = 12

XY = XA + AY = 16 + 16 = 32

The length of XY is 32.

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