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Question 1090481: P, Q, R, S and T are on line k such that Q is the midpoint of PT, R is the midpoint of QT, and S is the midpoint of RT. If PS = 9, then ehat is PT?
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website!
Step 1) Draw out a horizontal line and label it k.
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Step 2) Plot the points P, Q, R, S, T (in that order from left to right) on the line k
This is what you should have so far
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Step 3)
Since "S is the midpoint of RT", we know that RS = ST = x which is marked in purple for the figure below.
Since "R is the midpoint of QT", we know that QR = RT = y which is marked in green for the figure below.
Since "Q is the midpoint of PT", we know that PQ = QT = z which is marked in red for the figure below.
x, y and z are unknown for now
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Step 4)
Note how two copies of x can be combined to form y. On one hand, RS+ST = RT and at the same time x+x = 2x = y (in step 3 we have RS = ST = x; RT = y). This means y = 2x.
Put another way:
RT = RS + ST
RT = x + x
RT = 2x
RT = y
So y = 2x
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Step 5)
Using similar logic used in step 4, we can say that z = 2y. This is because z = QT and QT = QR+RT = y+y = 2y. By the transitive property, if z = QT and QT = 2y, then z = 2y.
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Step 6)
Refer to the figure in step 3 to determine that
PQ = z
QR = y
RS = x
Note how PQ+QR+RS = PS by the segment addition postulate.
So,
PS = PQ+QR+RS
PS = z+y+x
PS = 2y+y+x ... z has been replaced with 2y (see step 5)
PS = 2(2x)+2x+x ... every y has been replaced with 2x (see step 4)
PS = 4x+2x+x
PS = 7x
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Step 7)
Back in the previous step, we found out that PS = 7x. We are given that PS = 9. Use the substitution property to say
PS = 9
7x = 9
7x/7 = 9/7
x = 9/7
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Step 8)
Now we can find the length of PT
PT = PQ + QT
PT = z + z
PT = 2*z
PT = 2*(2y) ... z has been replaced with 2y (see step 5)
PT = 4*y
PT = 4*(2x) ... y has been replaced with 2x (see step 4)
PT = 8*x
PT = 8*(9/7) ... x has been replaced wth 9/7 (see step 7)
PT = 72/7
PT = 10.2857142857142
The decimal value is approximate. You can use long division or a calculator to come up with that result.
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Answer as a fraction: 72/7
Answer in decimal form: 10.2857142857142 (round this however you need to)
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