Author 
Message 
TAGS:

Hide Tags

Intern
Joined: 19 Oct 2009
Posts: 45

If vertices of a triangle have coordinates (1,0), (4,0), [#permalink]
Show Tags
Updated on: 27 Nov 2009, 10:41
Question Stats:
43% (01:23) correct 57% (01:20) wrong based on 760 sessions
HideShow timer Statistics
If vertices of a triangle have coordinates (1,0), (4,0), and (0,A) , is the area of the triangle greater than 15 ? (1) A < 3 (2) The triangle is right Hello friends, I solved this problem but I have a doubt. if we know that triangle is right triangle and hypotenus is 5, can we always safely assume other two sides as 3, 4 without knowing whether other sides are integer or not. Please help me i have my test 4 days away. Thanks.
Official Answer and Stats are available only to registered users. Register/ Login.
Originally posted by gmat620 on 27 Nov 2009, 08:46.
Last edited by gmat620 on 27 Nov 2009, 10:41, edited 1 time in total.



VP
Joined: 05 Mar 2008
Posts: 1427

Re: Urgent help required [#permalink]
Show Tags
27 Nov 2009, 09:47
gmat620 wrote: If vertices of a triangle have coordinates (1,0), (4,0), and (0,A) , is the area of the triangle greater than 15 ?
1. A < 3
2. The triangle is right
Hello friends, I solved this problem but I have a doubt. if we know that triangle is right triangle and hypotenus is 5, can we always safely assume other two sides as 3, 4 without knowing whether other sides are integer or not. Please help me i have my test 4 days away. Thanks. what if you have a 454590 triangle and the hypotenuse if 5 the other sides would be 5 sqrt2/2



Manager
Joined: 24 Sep 2009
Posts: 87

Re: Urgent help required [#permalink]
Show Tags
27 Nov 2009, 13:03
gmat620 wrote: If vertices of a triangle have coordinates (1,0), (4,0), and (0,A) , is the area of the triangle greater than 15 ?
1. A < 3
2. The triangle is right
Hello friends, I solved this problem but I have a doubt. if we know that triangle is right triangle and hypotenus is 5, can we always safely assume other two sides as 3, 4 without knowing whether other sides are integer or not. Please help me i have my test 4 days away. Thanks. You are right. Since the point (0,a) must be in 0yaxis, the hypotenus must be 5, hence the other two sides must be 3 and 4.
_________________
http://www.onlinestopwatch.com/ http://gmatsentencecorrection.blogspot.com/



Manager
Joined: 24 Sep 2009
Posts: 87

Re: Urgent help required [#permalink]
Show Tags
27 Nov 2009, 13:10
lagomez wrote: what if you have a 454590 triangle and the hypotenuse if 5
the other sides would be 5 sqrt2/2 this is true, but it's not the case here because the last vertex must be on y axis, so this is not a isosceles right triangle.
_________________
http://www.onlinestopwatch.com/ http://gmatsentencecorrection.blogspot.com/



Math Expert
Joined: 02 Sep 2009
Posts: 46207

Re: Urgent help required [#permalink]
Show Tags
27 Nov 2009, 13:45
gmat620 wrote: If vertices of a triangle have coordinates (1,0), (4,0), and (0,A) , is the area of the triangle greater than 15 ?
1. A < 3
2. The triangle is right
Hello friends, I solved this problem but I have a doubt. if we know that triangle is right triangle and hypotenus is 5, can we always safely assume other two sides as 3, 4 without knowing whether other sides are integer or not. Please help me i have my test 4 days away. Thanks. First of all right triangle with hypotenuse 5, doesn't mean that we have (3, 4, 5) right triangle. If we are told that values of all sides are integers, then yes: the only integer solution for right triangle with hypotenuse 5 would be (3, 4, 5). To check this: consider the right triangle with hypotenuse 5 inscribed in circle. We know that a right triangle inscribed in a circle must have its hypotenuse as the diameter of the circle. The reverse is also true: if the diameter of the circle is also the triangle’s hypotenuse, then that triangle is a right triangle. So ANY point on circumference of a circle with diameter \(5\) would make the right triangle with diameter. Not necessarily sides to be \(3\) and \(4\). For example we can have isosceles right triangle, which would be 454590: and the sides would be \(\frac{5}{\sqrt{2}}\). OR if we have 306090 triangle and hypotenuse is \(5\), sides would be \(2.5\) and \(2.5*\sqrt{3}\). Of course there could be many other combinations. Back to the original question: If vertices of a triangle have coordinates (1,0), (4,0), and (0,A) , is the area of the triangle greater than 15?(1) A < 3 > two vertices are on the Xaxis and the third vertex is on the Yaxis, below the point (0,3). The third vertex could be at (0,1) and the area would be less than 15 OR the third vertex could be at (0,100) and the area would be more than 15. So not sufficient. (2) The triangle is right. > Obviously as the third vertex is on the Yaxis, the right angle must be at the third vertex. Which means the hypotenuse is on Xaxis and equals to 5. Again if we consider the circle, the radius mus be 2.5 (half of the hypotenuse/diameter) and the third vertex must be one of two intersections of the circle with Yaxis. We'll get the two specific symmetric points for the third vertex, hence the area would be fixed and defined. Which means that it's possible to answer the question whether the area is more than 15, even not calculating actual value. Sufficient. Answer: B. If we want to know how the area could be calculated with the help of statement 2, here you go: One of the approaches: The equation of a circle is \((x  a)^2 + (yb)^2 = r^2\), where \((a,b)\) is the center and \(r\) is the radius. We know: \(r=2.5\), as the hypotenuse is 5. \(a=1.5\) and \(b=0\), as the center is on the Xaxis, at the point \((1.5, 0)\), half the way between the (1, 0) and (4, 0). We need to determine intersection of the circle with Yaxis, or the point \((0, y)\) for the circle. So we'll have \((01.5)^2 + (y0)^2 =2.5^2\) \(y^2=4\) > \(y=2\) and \(y=2\). The third vertex is either at the point \((0, 2)\) OR \((0,2)\). In any case \(Area=2*\frac{5}{2}=5\).
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread  All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
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. ,11 Mixed Questions, 12 Fresh Meat 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., 11 New DS set.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics



Intern
Joined: 19 Oct 2009
Posts: 45

Re: Urgent help required [#permalink]
Show Tags
27 Nov 2009, 15:47
Great explanation by Bunuel !! I wish Gmat wouldn't throw such questions on me.



Intern
Joined: 21 Aug 2009
Posts: 47

Re: Urgent help required [#permalink]
Show Tags
26 Sep 2010, 05:27
Hi Bunuel, Please tell me where I am going wrong. Calculated using the matrix formula to solve the area of the triangle. \(1/2 [1 (1A) + 4 (A1) +0(11)] >15\) \(A>7\) Option 1 says A< 3 Hence Statement A is sufficient. Am I missing something here?



Math Expert
Joined: 02 Sep 2009
Posts: 46207

Re: Urgent help required [#permalink]
Show Tags
26 Sep 2010, 06:32
ramgmat wrote: Hi Bunuel, Please tell me where I am going wrong. Calculated using the matrix formula to solve the area of the triangle. \(1/2 [1 (1A) + 4 (A1) +0(11)] >15\) \(A>7\) Option 1 says A< 3 Hence Statement A is sufficient. Am I missing something here? This is a valid approach if you are familiar with the formula which gives the area based on the coordinates of the three vertices of a triangle. If the vetices of a triangle are: \(A(a_x, a_y)\), \(B(b_x, b_y)\) and \(C(c_x,c_y)\) then the area of ABC is: \(area=\frac{a_x(b_yc_y)+b_x(c_ya_y)+c_x(a_yb_y)}{2}\). So if we consider: \(A(1,0)\), \(B(4,0)\), and \(C(0,A)\) then the area would be: \(area=\frac{1(0A)+4(A0)+0(00)}{2}\) > \(area=\frac{5A}{2}\). Question: is \(area=\frac{5A}{2}>15\) > is \(A>6\). Statement (1) says A>3, which is not sufficient to say whether \(A>6\). P.S. You made some errors in calculation and also didn't put the area formula in . Hope it helps.
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread  All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
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. ,11 Mixed Questions, 12 Fresh Meat 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., 11 New DS set.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics



Intern
Joined: 21 Aug 2009
Posts: 47

Re: Urgent help required [#permalink]
Show Tags
27 Sep 2010, 05:11
Thanks a lot Bunuel! I really appreciate this. I need to be careful of both my calculation as well as my silly mistakes



Manager
Joined: 25 Mar 2009
Posts: 52

Re: Urgent help required [#permalink]
Show Tags
28 Sep 2010, 22:32
Bunuel wrote: ramgmat wrote: Hi Bunuel, Please tell me where I am going wrong. Calculated using the matrix formula to solve the area of the triangle. \(1/2 [1 (1A) + 4 (A1) +0(11)] >15\) \(A>7\) Option 1 says A< 3 Hence Statement A is sufficient. Am I missing something here? This is a valid approach if you are familiar with the formula which gives the area based on the coordinates of the three vertices of a triangle. If the vetices of a triangle are: \(A(a_x, a_y)\), \(B(b_x, b_y)\) and \(C(c_x,c_y)\) then the area of ABC is: \(area=\frac{a_x(b_yc_y)+b_x(c_ya_y)+c_x(a_yb_y)}{2}\). So if we consider: \(A(1,0)\), \(B(4,0)\), and \(C(0,A)\) then the area would be: \(area=\frac{1(0A)+4(A0)+0(00)}{2}\) > \(area=\frac{5A}{2}\). Question: is \(area=\frac{5A}{2}>15\) > is \(A>6\). Statement (1) says A>3, which is not sufficient to say whether \(A>6\). P.S. You made some errors in calculation and also didn't put the area formula in . Hope it helps. \(area=\frac{a_x(b_yc_y)+b_x(c_ya_y)+c_x(a_yb_y)}{2}\). like this formulation, tks



Intern
Joined: 03 Sep 2010
Posts: 17

Re: Urgent help required [#permalink]
Show Tags
05 Oct 2010, 13:53
The answer is B As in (1) , A<3 , which means A = 3 , 100 , 200 anything > Not suffcient From (2) We conclude that the triangle is rt. angled at (0,A) We can set up the equation as (A0)/(0(1)) * (A0)/(04) = 1 as product of slopes of two perpendicular lines of a rt angled triangle is 1 A= +/ 2 which gives area of the rt angled traingle as 5 sq units < 15 sq units > sufficient Hence Answer is B Bunuel wrote: gmat620 wrote: If vertices of a triangle have coordinates (1,0), (4,0), and (0,A) , is the area of the triangle greater than 15 ?
1. A < 3
2. The triangle is right
Hello friends, I solved this problem but I have a doubt. if we know that triangle is right triangle and hypotenus is 5, can we always safely assume other two sides as 3, 4 without knowing whether other sides are integer or not. Please help me i have my test 4 days away. Thanks. First of all right triangle with hypotenuse 5, doesn't mean that we have (3, 4, 5) right triangle. If we are told that values of all sides are integers, then yes: the only integer solution for right triangle with hypotenuse 5 would be (3, 4, 5). To check this: consider the right triangle with hypotenuse 5 inscribed in circle. We know that a right triangle inscribed in a circle must have its hypotenuse as the diameter of the circle. The reverse is also true: if the diameter of the circle is also the triangle’s hypotenuse, then that triangle is a right triangle. So ANY point on circumference of a circle with diameter \(5\) would make the right triangle with diameter. Not necessarily sides to be \(3\) and \(4\). For example we can have isosceles right triangle, which would be 454590: and the sides would be \(\frac{5}{\sqrt{2}}\). OR if we have 306090 triangle and hypotenuse is \(5\), sides would be \(2.5\) and \(2.5*\sqrt{3}\). Of course there could be many other combinations. Back to the original question: If vertices of a triangle have coordinates (1,0), (4,0), and (0,A) , is the area of the triangle greater than 15?(1) A < 3 > two vertices are on the Xaxis and the third vertex is on the Yaxis, below the point (0,3). The third vertex could be at (0,1) and the area would be less than 15 OR the third vertex could be at (0,100) and the area would be more than 15. So not sufficient. (2) The triangle is right. > Obviously as the third vertex is on the Yaxis, the right angle must be at the third vertex. Which means the hypotenuse is on Xaxis and equals to 5. Again if we consider the circle, the radius mus be 2.5 (half of the hypotenuse/diameter) and the third vertex must be one of two intersections of the circle with Yaxis. We'll get the two specific symmetric points for the third vertex, hence the area would be fixed and defined. Which means that it's possible to answer the question whether the area is more than 15, even not calculating actual value. Sufficient. Answer: B. If we want to know how the area could be calculated with the help of statement 2, here you go: One of the approaches: The equation of a circle is \((x  a)^2 + (yb)^2 = r^2\), where \((a,b)\) is the center and \(r\) is the radius. We know: \(r=2.5\), as the hypotenuse is 5. \(a=1.5\) and \(b=0\), as the center is on the Xaxis, at the point \((1.5, 0)\), half the way between the (1, 0) and (4, 0). We need to determine intersection of the circle with Yaxis, or the point \((0, y)\) for the circle. So we'll have \((01.5)^2 + (y0)^2 =2.5^2\) \(y^2=4\) > \(y=2\) and \(y=2\). The third vertex is either at the point \((0, 2)\) OR \((0,2)\). In any case \(Area=2*\frac{5}{2}=5\).



Intern
Joined: 03 Sep 2010
Posts: 17

Re: Urgent help required [#permalink]
Show Tags
14 Oct 2010, 15:06
I appreciate the below solution but to solve in this lengthy method will need GMAC to provide 150 mins for Q instead of 75 mins
The concept to note here is product of slopes of 2 perpendicular lines = 1
Since the triangle can only be right angle at 0,A , then  A/4*A/1 = 1
Solve for A and then find out the max possible area of the triangle (B) is the answer
If you like my method , kindly provide comment



Manager
Joined: 24 Apr 2010
Posts: 56

Re: Urgent help required [#permalink]
Show Tags
15 Oct 2010, 05:48
Mikko wrote: pratikdas007 wrote: Hence Answer is B
As in (1) , A<3 , which means A = 3 , 100 , 200 anything > Not suffcient
From (2) We conclude that the triangle is rt. angled at (0,A)
We can set up the equation as (A0)/(0(1)) * (A0)/(04) = 1 as product of slopes of two perpendicular lines of a rt angled triangle is 1
A= +/ 2 which gives area of the rt angled traingle as 5 sq units < 15 sq units > sufficient
Hence Answer is B
i liked your method.... the bottom line seems to be from slope we can find coordinates of O,A hence find area and compare The value of A can be +ve or ve ....doesnt matter as it is length and because of coordinates of 2 given points which are in x axis we can say that right angle is not between them so seems like this method works... thanks



Manager
Joined: 19 Sep 2010
Posts: 156

Re: Urgent help required [#permalink]
Show Tags
15 Oct 2010, 09:36
Great explanation Bunuel...



Manager
Joined: 04 May 2009
Posts: 62
Location: Astoria, NYC

Re: Urgent help required [#permalink]
Show Tags
30 Nov 2010, 11:44
the biggest take away...We'll get the two specific symmetric points for the third vertex, hence the area would be fixed and defined. Which means that it's possible to answer the question whether the area is more than 15, even not calculating actual value.
Dont calculate....its a yes or no question. Fixed point means we can calculate area some how.



Manager
Joined: 30 Aug 2010
Posts: 89
Location: Bangalore, India

Re: Urgent help required [#permalink]
Show Tags
01 Dec 2010, 00:21
from the given info. the base of the triangle is 5 i.e the disantce between 1 and 4.
for the area to be greater than 15 , 0.5*5*hiegt > 15 ==> hiegt > 6
note that the third vertex(0,A) decides the hieght of the triangle.
Stmnt1. A < 3 ==> A could be 2 (hieght = 2 and answer to the question is NO) or 10 (hieght = 10 and the anser to the qtn is YES)...hence NOT suff.
stmnt2: the triangle is right
VERY IMPORTANT, USEFUL AND GMAT'S FAVORITE property: the angle on the semi circle is 90, means, if two vertices of the triangle are on the extreme sides of the diameter and the third is on the semi circle, then at the third vertex the angle is 90.
Using the above property and the stmnt 2 , we can say that the third verthex (0,A) is on a semi circle having the diameter connected between (1,0) and (4,0) ==> radius of the circle is 5 (distance b/w 1 and 4) / 2 = 2.5
hence the trianlge (right angle) in this semicircle would have the maximum area = 0.5 * 5 * 2.5 (height=radius) = 6.25 that is < 15 ==> the max area is < 15 ==> any other possible right angle triangle areas would be < 15 ==> answer to the question is "NO".
Answer "B".



Intern
Joined: 07 Sep 2010
Posts: 1

Re: Urgent help required [#permalink]
Show Tags
01 Dec 2010, 14:04
ok, this is simple. the area of our triangle is (!A!*5)/2 where 5 is a hypotenuse of the triangle (!14! = 5) first option is not sufficient, cuz !A! (absolute value of A) can be any number. 1, 10, 100000000 etc second option is sufficient. in any right angle triangle, the square of the altitude to the hypotenuse is equal to the product of two sectors it creates on hypotenuse. in our case one sector is 1 and the other is 4. so A*A=1*4, thus A=2. so the area is (2*5)/2. sorry my math english is not that good. tried my best to explain. Ans: B



Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 8102
Location: Pune, India

Re: Urgent help required [#permalink]
Show Tags
01 Dec 2010, 14:37
gmat620 wrote: If vertices of a triangle have coordinates (1,0), (4,0), and (0,A) , is the area of the triangle greater than 15 ?
1. A < 3
2. The triangle is right
Hello friends, I solved this problem but I have a doubt. if we know that triangle is right triangle and hypotenus is 5, can we always safely assume other two sides as 3, 4 without knowing whether other sides are integer or not. Please help me i have my test 4 days away. Thanks. First, your question  No, you cannot assume that the sides are 3 and 4 if you have 5 as hypotenuse. lagomez is right. What if it is 454590 triangle? The sides will not be 3 and 4. But in this question, you have something more. You know the line on which your third vertex of the triangle will fall. Attachment:
Ques2.jpg [ 9.29 KiB  Viewed 13821 times ]
There will be only 1 such right triangle so you will be able to say whether the area is greater than 15 or not. You don't need to find the triangle but you know this statement alone is sufficient.
_________________
Karishma Veritas Prep  GMAT Instructor My Blog
Get started with Veritas Prep GMAT On Demand for $199
Veritas Prep Reviews



Math Expert
Joined: 02 Sep 2009
Posts: 46207

Re: Urgent help required [#permalink]
Show Tags
01 Dec 2010, 14:50
VeritasPrepKarishma wrote: gmat620 wrote: If vertices of a triangle have coordinates (1,0), (4,0), and (0,A) , is the area of the triangle greater than 15 ?
1. A < 3
2. The triangle is right
Hello friends, I solved this problem but I have a doubt. if we know that triangle is right triangle and hypotenus is 5, can we always safely assume other two sides as 3, 4 without knowing whether other sides are integer or not. Please help me i have my test 4 days away. Thanks. First, your question  No, you cannot assume that the sides are 3 and 4 if you have 5 as hypotenuse. lagomez is right. What if it is 454590 triangle? The sides will not be 3 and 4. But in this question, you have something more. You know the line on which your third vertex of the triangle will fall. Attachment: Ques2.jpg There will be only 1 such right triangle so you will be able to say whether the area is greater than 15 or not. You don't need to find the triangle but you know this statement alone is sufficient. Little correction here: actually there will be 2 such right triangles, the second one will be the mirror image of the first ( urgenthelprequired87344.html#p656628).
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread  All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
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. ,11 Mixed Questions, 12 Fresh Meat 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., 11 New DS set.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics



Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 8102
Location: Pune, India

Re: Urgent help required [#permalink]
Show Tags
01 Dec 2010, 15:02
Yes, that's true. But since we are only concerned with area, we can pretty much ignore the mirror images for this statement.
_________________
Karishma Veritas Prep  GMAT Instructor My Blog
Get started with Veritas Prep GMAT On Demand for $199
Veritas Prep Reviews




Re: Urgent help required
[#permalink]
01 Dec 2010, 15:02



Go to page
1 2 3
Next
[ 55 posts ]



