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If P and Q are two points on the line 3x + 4y =-15 such that

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If P and Q are two points on the line 3x + 4y =-15 such that [#permalink] New post 21 Oct 2010, 06:47
If P and Q are two points on the line 3x + 4y = -15 such that OP and OQ= 9 units, The area of triangle POQ is given by

A. 18* sqrt(2)
B. 3* sqrt(2)
C. 6* sqrt(2)
D. 15*sqrt(2)
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Re: Good Question ...Number system [#permalink] New post 21 Oct 2010, 17:22
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kobinaot wrote:
help pls!

2)If P and Q are two points on the line 3x + 4y = -15 such that OP and OQ= 9 units, The area of triangle POQ is given by

a) 18* sqrt(2)
b) 3* sqrt(2)
c) 6* sqrt(2)
d) 15*sqrt(2)


Look at the the diagram below:
Attachment:
graph.php.png
graph.php.png [ 17.5 KiB | Viewed 1476 times ]

OC is perpendicular to AB (PQ), so it's height of AOB and POQ;

OA=5 and OB=\frac{15}{4} (points A and B are intersection of the line 3x+4y=-15 with the X and Y axis respectively) --> AB=hypotenuse=\sqrt{OA^2+OB^2}=\frac{25}{4};

As AOB and OCB are similar then \frac{OC}{OB}=\frac{OA}{AB} --> OC=3;

CP=CQ=\sqrt{OP^2-OC^2}=\sqrt{9^2-3^2}=6\sqrt{2};

PQ=CP+CQ=12\sqrt{2};

Area_{OPQ}=\frac{base*height}{2}=\frac{PQ*OC}{2}=\frac{12\sqrt{2}*3}{2}=18\sqrt{2}.

Answer: A.

P.S. Please post the new questions in separate threads.
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Re: The area of triangle POQ [#permalink] New post 21 Oct 2010, 18:02
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another way to calculate OC is:
(distance of (a,b) from mx+ny+z=0 is: (ma+nb+z)/sqrt(m^2 + n^2)
SO, OC = distance of O from the line 3x+4y+15= 0 is:

(3 * 0 + 4*0 +15)/sqrt(3^2 + 4^2) = 15/5 = 3
Then we calculate CP=CQ as per Bunel.. and the answer comes out as a) 18* sqrt(2)
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Re: Good Question ...Number system [#permalink] New post 22 Oct 2010, 22:08
Bunuel wrote:
kobinaot wrote:
help pls!

2)If P and Q are two points on the line 3x + 4y = -15 such that OP and OQ= 9 units, The area of triangle POQ is given by

a) 18* sqrt(2)
b) 3* sqrt(2)
c) 6* sqrt(2)
d) 15*sqrt(2)


Look at the the diagram below:
Attachment:
graph.php.png

OC is perpendicular to AB (PQ), so it's height of AOB and POQ;

OA=5 and OB=\frac{15}{4} (points A and B are intersection of the line 3x+4y=-15 with the X and Y axis respectively) --> AB=hypotenuse=\sqrt{OA^2+OB^2}=\frac{25}{4};

As AOB and OCB are similar then \frac{OC}{OB}=\frac{OA}{AB} --> OC=3;

CP=CQ=\sqrt{OP^2-OC^2}=\sqrt{9^2-3^2}=6\sqrt{2};

PQ=CP+CQ=12\sqrt{2};

Area_{OPQ}=\frac{base*height}{2}=\frac{PQ*OC}{2}=\frac{12\sqrt{2}*3}{2}=18\sqrt{2}.

Answer: A.

P.S. Please post the new questions in separate threads.



Bunuel, what makes us think that point o is Origin in this case???
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Re: Good Question ...Number system [#permalink] New post 23 Oct 2010, 05:35
utin wrote:
Bunuel, what makes us think that point o is Origin in this case???


Well, we must know the coordinates of the point O to solve the question. As origin in coordinate geometry usually marked with letter O (as the center of a circle) then we can assume that O is origin in this case. Though I think on GMAT it would be explicitly stated.
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Re: Good Question ...Number system [#permalink] New post 23 Oct 2010, 21:02
Bunuel wrote:
utin wrote:
Bunuel, what makes us think that point o is Origin in this case???


Well, we must know the coordinates of the point O to solve the question. As origin in coordinate geometry usually marked with letter O (as the center of a circle) then we can assume that O is origin in this case. Though I think on GMAT it would be explicitly stated.


Thanks for the info...Bunuel :)
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Re: Good Question ...Number system [#permalink] New post 13 Jul 2012, 04:30
Bunuel wrote:
kobinaot wrote:
help pls!

2)If P and Q are two points on the line 3x + 4y = -15 such that OP and OQ= 9 units, The area of triangle POQ is given by

a) 18* sqrt(2)
b) 3* sqrt(2)
c) 6* sqrt(2)
d) 15*sqrt(2)


Look at the the diagram below:
Attachment:
graph.php.png

OC is perpendicular to AB (PQ), so it's height of AOB and POQ;

OA=5 and OB=\frac{15}{4} (points A and B are intersection of the line 3x+4y=-15 with the X and Y axis respectively) --> AB=hypotenuse=\sqrt{OA^2+OB^2}=\frac{25}{4};

As AOB and OCB are similar then \frac{OC}{OB}=\frac{OA}{AB} --> OC=3;

CP=CQ=\sqrt{OP^2-OC^2}=\sqrt{9^2-3^2}=6\sqrt{2};

PQ=CP+CQ=12\sqrt{2};

Area_{OPQ}=\frac{base*height}{2}=\frac{PQ*OC}{2}=\frac{12\sqrt{2}*3}{2}=18\sqrt{2}.

Answer: A.

P.S. Please post the new questions in separate threads.


But how can we assume that O is the origin??
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Re: Good Question ...Number system [#permalink] New post 13 Jul 2012, 04:34
MacFauz wrote:
Bunuel wrote:
kobinaot wrote:
help pls!

2)If P and Q are two points on the line 3x + 4y = -15 such that OP and OQ= 9 units, The area of triangle POQ is given by

a) 18* sqrt(2)
b) 3* sqrt(2)
c) 6* sqrt(2)
d) 15*sqrt(2)


Look at the the diagram below:
Attachment:
graph.php.png

OC is perpendicular to AB (PQ), so it's height of AOB and POQ;

OA=5 and OB=\frac{15}{4} (points A and B are intersection of the line 3x+4y=-15 with the X and Y axis respectively) --> AB=hypotenuse=\sqrt{OA^2+OB^2}=\frac{25}{4};

As AOB and OCB are similar then \frac{OC}{OB}=\frac{OA}{AB} --> OC=3;

CP=CQ=\sqrt{OP^2-OC^2}=\sqrt{9^2-3^2}=6\sqrt{2};

PQ=CP+CQ=12\sqrt{2};

Area_{OPQ}=\frac{base*height}{2}=\frac{PQ*OC}{2}=\frac{12\sqrt{2}*3}{2}=18\sqrt{2}.

Answer: A.

P.S. Please post the new questions in separate threads.


But how can we assume that O is the origin??


Please read the thread completely: if-p-and-q-are-two-points-on-the-line-3x-4y-15-such-that-103360.html#p805599
_________________

PLEASE READ AND FOLLOW: 11 Rules for Posting!!!

RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory

COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. NEW!!!

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS ; 9 Devil's Dozen!!!; 10 Number Properties set. NEW!!!


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Re: Good Question ...Number system   [#permalink] 13 Jul 2012, 04:34
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