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The angle made by the line root{3}x+y+6=0, with yaxis is
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12 Dec 2010, 08:26
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The angle made by the line \(\sqrt{3}x+y+6=0\), with yaxis is A. 60 degrees B. 30 degrees C. 45 degrees D. 120 degrees E. None of the above
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Re: Slope Angle
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12 Dec 2010, 08:54
mailnavin1 wrote: The angle made by the line \sqrt{3}x+y+6=0, with yaxis is
1. 60 degrees 2. 30 degrees 3. 45 degrees 4. 120 degrees 5. None of the above We have the line \(y=\sqrt{3}*x6\). Attachment:
untitled.PNG [ 14.87 KiB  Viewed 10682 times ]
Find the xintercept of the line (xintercept of a line the value of x for y=0, the point where the line crosses Xaxis): y=0 > \(x=2\sqrt{3}\); Find the yintercept of the line (yintercept of a line the value of y for x=0, the point where the line crosses Yaxis): x=0 > \(y=6\); In the right triangle created by the given line and the axis the ratio of lengths of the legs is \(2\sqrt{3}:6=1:\sqrt{3}\) so we have 30°, 60°, and 90° right triangle (in a right triangle where the angles are 30°, 60°, and 90° the sides are always in the ratio \(1 : \sqrt{3}: 2\). Notice that the smallest side (1) is opposite the smallest angle (30°), and the longest side (2) is opposite the largest angle (90°)). Now, the angle at yaxis is opposite the smallest side \(2\sqrt{3}\) so it equals to 30°. Answer: B. Or you can notice that as the slope, which is th ratio of the rise over run is equal to \(\sqrt{3}\) then the ratio of the length of the leg at yaxis over the length of the leg at xaxis will be \(\sqrt{3}\), which again tells us that we have 30°, 60°, and 90° right triangle. For more on this check Triangles and Coordinate Geometry chapters of Math Book: mathtriangles87197.html and mathcoordinategeometry87652.htmlHope it's clear.
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Re: Slope Angle
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12 Dec 2010, 09:47
that was smart. I tried to use m = tan(degree), Is this wrong. If no can you please explain how to compute using this.
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Re: Slope Angle
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12 Dec 2010, 10:33
mailnavin1 wrote: that was smart. I tried to use m = tan(degree), Is this wrong. If no can you please explain how to compute using this. Every GMAT geometry question can be solved without trigonometry, so you don't need trigonometry for GMAT. But yes, \(slope=m=tan(\alpha)\), where \(\alpha\) is the angle line makes with positive xaxis, so as \(m=\sqrt{3}=tan(\alpha)\) > \(\alpha=120^{\circ}\) > angle at yaxis will be 12090=30.
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Re: Slope Angle
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13 Dec 2010, 08:26
Good explanation. Thanks Bunuel
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Re: Slope Angle
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15 Dec 2010, 23:03
This question is incomplete, for answer to be 30 degrees, question should ask the smaller angle between line and yaxis. Otherwise answer can be either 150 degrees or 30 degrees.
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Re: Slope Angle
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16 Dec 2010, 05:28
hermit84 wrote: This question is incomplete, for answer to be 30 degrees, question should ask the smaller angle between line and yaxis. Otherwise answer can be either 150 degrees or 30 degrees. Though I see your point, still only one answer from 150 and 30 is listed among answer choices, so we can safely choose 30 as correct one.
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Re: Slope Angle
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14 Jan 2011, 17:35
Bunuel wrote: mailnavin1 wrote: The angle made by the line \sqrt{3}x+y+6=0, with yaxis is
1. 60 degrees 2. 30 degrees 3. 45 degrees 4. 120 degrees 5. None of the above We have the line \(y=\sqrt{3}*x6\). Attachment: untitled.PNG Find the xintercept of the line (xintercept of a line the value of x for y=0, the point where the line crosses Xaxis): y=0 > \(x=2\sqrt{3}\); Find the yintercept of the line (yintercept of a line the value of y for x=0, the point where the line crosses Yaxis): x=0 > \(y=6\); In the right triangle created by the given line and the axis the ratio of lengths of the legs is \(2\sqrt{3}:6=1:\sqrt{3}\) so we have 30°, 60°, and 90° right triangle (in a right triangle where the angles are 30°, 60°, and 90° the sides are always in the ratio \(1 : \sqrt{3}: 2\). Notice that the smallest side (1) is opposite the smallest angle (30°), and the longest side (2) is opposite the largest angle (90°)). Now, the angle at yaxis is opposite the smallest side \(2\sqrt{3}\) so it equals to 30°. Answer: B. Or you can notice that as the slope, which is th ratio of the rise over run is equal to \(\sqrt{3}\) then the ratio of the length of the leg at yaxis over the length of the leg at xaxis will be \(\sqrt{3}\), which again tells us that we have 30°, 60°, and 90° right triangle. For more on this check Triangles and Coordinate Geometry chapters of Math Book: mathtriangles87197.html and mathcoordinategeometry87652.htmlHope it's clear. Hello Bunnel, Can you pls explain how did draw the graph... How did you take the points of x and y..



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Re: Slope Angle
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14 Jan 2011, 18:25
jullysabat wrote: Hello Bunnel,
Can you pls explain how did draw the graph... How did you take the points of x and y.. \(y=mx+b\) is called pointintercept form of equation of a line (in our case it's \(y=\sqrt{3}*x6\)), where \(m\) is the slope of the line; \(b\) is the yintercept of the line; \(x\) is the independent variable of the function \(y\). To draw a graph of a line you need two points (any two distinct points (x,y) which satisfy the equation of a line) and then you just draw a line passing these points. Usually X and Y intercepts are best choices for the points. Xintercept is the point where a line (a graph) crosses the xaxis. So it's the point on xaxis, any point on xaxis has ycoordinate equal to zero, which means that Xintercept is the point \((x,0)\)  the value of \(x\) when \(y=0\): \(y=mx+b\) > \(0=mx+b\) > \(x=\frac{b}{m}\). So Xintercept of a line \(y=mx+b\) is \(x=\frac{b}{m}\); Yintercept is the point where a line (a graph) crosses the yaxis. So it's the point on yaxis, any point on yaxis has xcoordinate equal to zero, which means that Yintercept is the point \((0,y)\)  the value of \(y\) when \(x=0\): \(y=mx+b\) > \(y=m*0+b\) > \(y=b\). So Yintercept of a line \(y=mx+b\) is \(y=b\). In our case for the line given by \(y=\sqrt{3}*x6\): Xintercept will be \((2\sqrt{3}, \ 0)\), as xintercept of a line is the value of x for y=0, the point where the line crosses Xaxis: y=0 > \(x=2\sqrt{3}\); Yintercept will be \((0, \ 6)\), as yintercept of a line is the value of y for x=0, the point where the line crosses Yaxis: x=0 > \(y=6\). Check Coordinate Geometry chapter for more: mathcoordinategeometry87652.htmlHope it's clear.
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Re: Slope Angle
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14 Jan 2011, 19:05
Bunuel wrote: jullysabat wrote: Hello Bunnel,
Can you pls explain how did draw the graph... How did you take the points of x and y.. \(y=mx+b\) is called pointintercept form of equation of a line (in our case it's \(y=\sqrt{3}*x6\)), where \(m\) is the slope of the line; \(b\) is the yintercept of the line; \(x\) is the independent variable of the function \(y\). To draw a graph of a line you need two points (any two distinct points (x,y) which satisfy the equation of a line) and then you just draw a line passing these points. Usually X and Y intercepts are best choices for the points. Xintercept is the point where a line (a graph) crosses the xaxis. So it's the point on xaxis, any point on xaxis has ycoordinate equal to zero, which means that Xintercept is the point \((x,0)\)  the value of \(x\) when \(y=0\): \(y=mx+b\) > \(0=mx+b\) > \(x=\frac{b}{m}\). So Xintercept of a line \(y=mx+b\) is \(x=\frac{b}{m}\); Yintercept is the point where a line (a graph) crosses the yaxis. So it's the point on yaxis, any point on yaxis has xcoordinate equal to zero, which means that Yintercept is the point \((0,y)\)  the value of \(y\) when \(x=0\): \(y=mx+b\) > \(y=m*0+b\) > \(y=b\). So Yintercept of a line \(y=mx+b\) is \(y=b\). In our case for the line given by \(y=\sqrt{3}*x6\): Xintercept will be \((2\sqrt{3}, \ 0)\), as xintercept of a line is the value of x for y=0, the point where the line crosses Xaxis: y=0 > \(x=2\sqrt{3}\); Yintercept will be \((0, \ 6)\), as yintercept of a line is the value of y for x=0, the point where the line crosses Yaxis: x=0 > \(y=6\). Check Coordinate Geometry chapter for more: mathcoordinategeometry87652.htmlHope it's clear. Yeah its clear now... Thanks..



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Re: Slope Angle
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16 Jan 2011, 13:37
very nice work Bunuel. thanks for putting so much effort.
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Re: Slope Angle
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10 Mar 2011, 02:58
But why the tan(@) formula works on this. As Y axis has a slope = infinity. So this formula doesn't work and i had chosen last option which says "None of the above". I know this option may not be there in GMAT but why can't we apply a particular formula to reach the same result. Tan(@) = (m1  m2)/(1+m1*m2)
Where m1 and m2 are the slopes of two lines and "@" is angle between these line.



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Re: Slope Angle
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22 Apr 2013, 06:37
Bunuel wrote: mailnavin1 wrote: The angle made by the line \sqrt{3}x+y+6=0, with yaxis is
1. 60 degrees 2. 30 degrees 3. 45 degrees 4. 120 degrees 5. None of the above Or you can notice that as the slope, which is th ratio of the rise over run is equal to \(\sqrt{3}\) then the ratio of the length of the leg at yaxis over the length of the leg at xaxis will be \(\sqrt{3}\), which again tells us that we have 30°, 60°, and 90° right triangle. Hope it's clear. Hi Bunuel , Sorry to dig in an old post . I am unable to follow your second approach properly (i.e how did you find the ratio length of y to length of x as sqrt3) . Could you please elaborate it more . thanks !



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Re: Slope Angle
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22 Apr 2013, 22:34
Bunuel wrote: \(2\sqrt{3}:6=1:\sqrt{3}\)
Hope it's clear. The explanation of this question is fantastic and one can quickly solve these kind of questions after understanding it. The only concept that i am not able to understand is how \(2\sqrt{3}:6\) turn into \(1:\sqrt{3}\) I will really appreciate if anyone can help me in understanding this concept. Thanks & Regards Vinni



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Re: Slope Angle
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01 Sep 2013, 08:18
Vini,
A line of the form y=mx+b, has a slope 'm'
Convert the given equation in this form to get y=rt3x 6.
Since quesn deals in slope you need not calculate intercepts, clearly slope is negative and is tan(rt30 which is 60 degree. It lies in 3rd quadrant Since slope is defined as what angle a line makes with x axis, the angle with y axis is 9060 = 30.
If you want to know about intercepts, the above eqn cut x axis when y=0 rt? So when y=0, x = 6/rt3 = 3* 2 / rt3 = 2rt3. Also when x=0, y=6.
Hope it helped.



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Re: The angle made by the line root{3}x+y+6=0, with yaxis is
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12 Oct 2013, 06:40
Also for a 454590 angled right angled triangle the corresponding sides are in the ratio 1: 1: \(\sqrt{2}\)



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Re: The angle made by the line root{3}x+y+6=0, with yaxis is
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12 Oct 2013, 07:01
"The explanation of this question is fantastic and one can quickly solve these kind of questions after understanding it. The only concept that i am not able to understand is how 2\(\sqrt{3}\):6 turn into 1:\(\sqrt{3}\)"
here goes first cancel with 2 on either side of the ratio \(\sqrt{3}\) : 3 on the right 3 can be written as \(\sqrt{3}\)*\(\sqrt{3}\) \(\sqrt{3}\) : \(\sqrt{3}\)* \(\sqrt{3}\) cancel \(\sqrt{3}\) you get 1: \(\sqrt{3}\)



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Re: The angle made by the line root{3}x+y+6=0, with yaxis is
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12 Nov 2013, 04:07
Another idea: Observe buneul's diagram. The coordinates on the ve xaxis and ve yaxis are (2\(\sqrt{3}\),0) and (0,6). Now find the distance between them using the formula \(\sqrt{x^2+y^2}\). We get 4\(\sqrt{3}\)=2*2\(\sqrt{3}\). Now we know that the ratio of sides for a 306090 triangle is x:\(\sqrt{3}\)x:2x. From distance calculation above we can see that the following ratio holds perfectly good 30:60:90=2\(\sqrt{3}\): \(\sqrt{3}\)*(2\(\sqrt{3}\)): 2*(2\(\sqrt{3}\)). Comparing we can say that side opposite to 2\(\sqrt{3}\) measures 30 degrees.



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Re: The angle made by the line root{3}x+y+6=0, with yaxis is
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11 Apr 2016, 18:40
mailnavin1 wrote: The angle made by the line \(\sqrt{3}x+y+6=0\), with yaxis is
A. 60 degrees B. 30 degrees C. 45 degrees D. 120 degrees E. None of the above this one can be solved easily just by picturing the graph... rewrite equation: y=  sqrt(3)*x 6 when x=0, y=6. when x=sqrt(3), y=9 so 3 down the y axis, and 1 to the right of x axis. we can draw a right triangle, in which 2 legs are known. sqrt(3) & 3. now..hypotenuse^2 is 3^2 + sqrt(3)^2 = 12. so hypotenuse is 2*sqrt(3) clearly not 45 and 120 we know that sides of a triangle with angles 306090 must be in this proportion: x:x*sqrt(3):2x. x=sqrt(3); 2x=2*sqrt(3), and x*sqrt(3) = 3. we have a 306090 triangle, and the angle formed at the intersection with y axis is 30 degrees.



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