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In the figure above, segment PT has length 36, Q is the midpoint of PR

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Bunuel
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In the figure above, segment PT has length 36, Q is the midpoint of PR [#permalink] New post   12 Jul 2017, 01:37
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In the figure above, segment PT has length 36, Q is the midpoint of PR, R is the midpoint of QS, and S is the midpoint of QT. What is the length of segment QS?

(A) 7.2
(B) 12.2
(C) 12.4
(D) 14.2
(E) 14.4


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generis
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In the figure above, segment PT has length 36, Q is the midpoint of PR [#permalink] New post   Updated on: 12 Jul 2017, 05:31
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Bunuel wrote:

In the figure above, segment PT has length 36, Q is the midpoint of PR, R is the midpoint of QS, and S is the midpoint of QT. What is the length of segment QS?

(A) 7.2
(B) 12.2
(C) 12.4
(D) 14.2
(E) 14.4


Show Spoiler
Attachment:
2017-07-12_1235.png

By definition of midpoint:

PQ = QR
QR = RS, and therefore
PQ = RS

Those are three segments of equal length. Call their length x. Distance between P to Q, from Q to R, and from R to S is x. Or PQ = QR = RS = x

Also by definition of midpoint, QS = ST

QS = (QR + RS) = (x + x) = 2x and
QS = ST, so
ST = QR + RS = 2x

To the left of point S, we have x + x + x (P -> Q -> R) = 3x

To the right of point S = ST, which from above is 2x.

3x + 2x = 5x
5x = 36
x = 7.2

From above, QS = 2x = 14.4

Answer E

Originally posted by generis on 12 Jul 2017, 05:20.
Last edited by generis on 12 Jul 2017, 05:31, edited 3 times in total.
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Re: In the figure above, segment PT has length 36, Q is the midpoint of PR [#permalink] New post   12 Jul 2017, 05:22
Bunuel wrote:

In the figure above, segment PT has length 36, Q is the midpoint of PR, R is the midpoint of QS, and S is the midpoint of QT. What is the length of segment QS?

(A) 7.2
(B) 12.2
(C) 12.4
(D) 14.2
(E) 14.4


Show Spoiler
Attachment:
2017-07-12_1235.png



It should be E.

If we can mark PQ = x, QR is also x, similarly, RS is also S, but ST becomes 2x. This may not be easy to comprehend initially but just drawing on paper and working your way through it should help you understand it fairly quickly.

5x = 36
x = 7.2

Since QS is 2 times x, it should be 14.4.
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In the figure above, segment PT has length 36, Q is the midpoint of PR [#permalink] New post   15 Jul 2017, 02:01
Let PQ=a QR= b RS=c and ST= d then PT = a+b+c+d=36 and
since q is the midpoint of PR we get PQ=QR so a=b since R is the midpoint of QS QR= RS so b=c since S is the midpoint of QT= QR+RS+ST also QR+RS=ST so b+c=d since a=b b=c therefore a=c 2c=d
36=5c c=7.2. b=c=7.2
Therefore b+c = 14.4 (E)
Hope this method helps.
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Re: In the figure above, segment PT has length 36, Q is the midpoint of PR [#permalink] New post   17 Jul 2017, 10:34
Bunuel wrote:

In the figure above, segment PT has length 36, Q is the midpoint of PR, R is the midpoint of QS, and S is the midpoint of QT. What is the length of segment QS?

(A) 7.2
(B) 12.2
(C) 12.4
(D) 14.2
(E) 14.4


Show Spoiler
Attachment:
2017-07-12_1235.png



ANSWER:
=====

PT = PQ+QR+RS+ST

We know that PQ=QR AND QR=RS AND QS=ST
NOTICE QS=2QR=2PQ.

SUBSTITUTE THESE VALUES IN EQUATION
PT = PQ+QR+RS+ST=36
PQ+PQ+PQ+2PQ=36
5PQ=36
PQ=36/5=7.2

QS =2PQ =2*7.2 = 14.4
HENCE, THE ANSWER IS E.
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Re: In the figure above, segment PT has length 36, Q is the midpoint of PR [#permalink]
17 Jul 2017, 10:34
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