Points 'n Lines

Geometry Level 1

Points Q and R are on segment PS, which has length 24. If PQ is half as long as QS, and QR is 3 times as long as RS, how long is RS?


The answer is 4.

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Aly Ramy by Aly Ramy , 22, Egypt

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12 solutions

Pallavi Singhal
Sep 12, 2014

if PQ=x then QS=2x thus x+2x=24, --> x =8 thus QS=16 now RS=x, QR=3x thus x+3x=16 this implies rs=4

2 Helpful 0 Interesting 0 Brilliant 0 Confused

PS = PQ+QR+RS = 24, substitute QR with 3RS, PQ +4RS = 24, from PQ = QS/2, PQ =( QR + RS)/2, PQ = (3RS +RS)/2, PQ = 2RS therefore PS = 2RS = 4RS = 24, simplify: RS = 4

Sards Sarmiento - 6 years, 8 months ago
Shereen Monica
Sep 11, 2014

PQ+QR+RS = 24

Substituting the given values in terms of RS (the required distance),

0.5(3RS+RS) + 3RS + RS = 24

We get, 6RS = 24

So, RS = 4

1 Helpful 0 Interesting 0 Brilliant 0 Confused

PQ=x, QR=y, RS=z

x+y+z=24 x=(y+z)/2 y=3z

z=4, y=12, x=8

Carlos DiGio - 6 years, 9 months ago
Seif Eldin Adel
Sep 20, 2014

Let RS = x So , QR =3x QS = QR + RS = 4x PQ = QS/2 = 2x PS = 2x + 3x + x = 6x so , 6x=24 x=4

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Samrat Hasan
Sep 19, 2014

If RS=x then QR=3x thus QS=4x and PQ=2x Now ATQ 4x+2x=24 => x =4..... thus RS=4

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Ryan Evan Estaño
Sep 16, 2014

PS=24 PQ=1/2 (QS) QR=3RS

QS= QR + RS = 3RS + 1RS = 4RS PQ= 1/2(QS) = 1/2 (4RS) = 2RS PS= PQ + QS = 4RS + 2RS = 6RS 6RS = 24 RS= 4

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Azhar T
Sep 16, 2014

PS=24( ie PQ+QR+RS=24) given QR=3RS so, PQ+4RS=24 given 2PQ=QS(QS=3RS+RS) so, PQ=2RS 2RS+4RS=24 so RS =4

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Antonio Fanari
Sep 16, 2014

|PS| = 24; |PQ| = |QS|/2; |QR| = 3|QS|; |RS| = ? (1)

|PQ| + |QR| + |RS| = |PS|; (2)

|QR| + |RS| = |QS|; (3)

|QS|/2 + |QS| = |PS|;

3|QS|/2 = |PS|;

|QS| = 2|PS|/3; (4)

3|RS| + |RS| = |QS| = 2|PS|/3;

4|RS| = 2|PS|/3;

|RS| = |PS|/6 = 24/6 = 4.

0 Helpful 0 Interesting 0 Brilliant 0 Confused
V Vishnu Prasad
Sep 12, 2014

PQ+QR+RS = 24 (or) PQ+QS =24

given PQ=0.5 QS...Substitute in above->> 1.5QS = 24.....>>QS=16

QR+RS= QS=16....>> QR=3 RS(Given)...>> 3RS+RS = 16.....>>4RS=16.... SO, RS= 4

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Auggie Alcala
Sep 12, 2014

Wow all these other answers look complicated, I just did it in my head and got 4.

0 Helpful 0 Interesting 0 Brilliant 0 Confused
M. H.
Sep 11, 2014

PS=24.

PS = 24 = PQ + QS = PQ + 2(PQ). 3(PQ) = 24. PQ = 8. QS = 16.

QS = 16 = QR + RS = 3(RS) + RS = 4(RS). RS = 4.

0 Helpful 0 Interesting 0 Brilliant 0 Confused
K R
Sep 11, 2014

PQ=X, QS=2X, PQ+PS= 3X= 24, so X= 8. So QS=16. QR= 3Z, RS= 1Z, QR+RS= 3Z+Z= 4Z= QS=16, so Z= 4. So RS=4

0 Helpful 0 Interesting 0 Brilliant 0 Confused
John Wilcox
Sep 11, 2014

PQ+QS=24, (QS/2)+QS=24, (3QS/2)=24, QS=16, QR+RS=QS, QR+RS=16, QR+(QR/3)=16, (4QR/3)=16, QR=12, RS =4

0 Helpful 0 Interesting 0 Brilliant 0 Confused

PS=24

let X = RS,

3X = QR,

2X = PQ ......(PQ=(QR+QS)/2)

2X+3X+X=24......then simplify

6X=24...divide both sides by 6

X=4

Sheila Palagtiw - 6 years, 9 months ago

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