M08-15

Question

What is the degree measure of angle  a  (where  a  is less than  90  degrees) formed by the intersection of two lines with equations  y=mx+b  and  x=y+bm ?

(1)  m=2 
 
(2)  m=b

Bunuel · Answer

Official Solution:

What is the degree measure of angle  a  (where  a  is less than  90  degrees) formed by the intersection of two lines with equations  y=mx+b  and  x=y+bm ? 
 
To find the angle between any two lines, we need to determine their slopes. The slope of a line describes its  steepness  and  direction , hence knowing the slopes of two lines allows us to calculate the angle between them. Thus, to determine the angle between the lines with equations  y=mx+b  and  y=x-bm , we must first find the slope of each line. The slope of the first line is given by  m  and the slope of the second line is 1. Therefore, the value of  m  is the only unknown needed to calculate the angle between the lines.
 
Note here, that when tackling a GMAT data sufficiency question that asks for the value of something, your main objective is not to solve for the actual value. Instead, you need to determine whether the information given in the statements is sufficient to calculate the value. In other words, the goal is to assess whether you have enough data to arrive at a single, unambiguous answer. Obtaining a numerical result is not always required to achieve this goal, as your main focus should be on evaluating the given information and determining its adequacy to solve the problem.
 
(1)  m=2 . Sufficient.
 
(2)  m=b . Irrelevant information. Not sufficient.

Answer: A

Mrtinhnv · Answer

Dear Bunuel!
I can find the slope of the first line is m and the slope of the second line is 1
But I don’t know How can we find value of a if we know the slope of two lines. You can more detail.
Thanks so much!

askhere · Answer

Slope of a line is a measure of its inclination with respect to the horizontal. 
The more the inclination the more will be the slope, lesser the inclination less will be the slope.
In general, slope of a line is defined as m = tan θ , where θ is the angle between the line and the horizontal axis.
(Even though trigonometry is not involved in GMAT, if you understand the concept of slope it will help you in solving 
problems such as these. In fact the formula for slope m = (y 2 -y 1 )/(x 2 -x 1 ) is actually derived from m = tan θ)

So, if you know slope of a line, you know the angle with which its is inclined.   
Since the given problem is a data sufficiency problem you dont have to solve for anything. You just need to say whether the info given is sufficient to
answer the question.

See attached figure
From the question stem you know the slope of one line which means the angle at which this line is inclined (θ 1 )
From the statement 1 you know the slope of other line which means the angle at which this line is inclined (θ 2 )
The difference between these two angles will give you the angle at which they are intersected.

atl12688 · Answer

I'm struggling to see how Slope of line two equals 1. Can anyone explain that part with some added detail, thank you!

Bunuel · Answer

If the equation of the line is given in the Point-intercept form: y=mx+b, then m is the slope. Therefore, the slope of y=x−bm, is 1 (coefficient of x). Check for more here: math-coordinate-geometry-87652.html Hope it helps.

matmen · Answer

I think this is a  high-quality  question.

GDT · Answer

VeritasKarishma ,  chetan2u

Can you pls explain how knowing that the slope is 1 & 2 here, helps us in finding a

Thanks in advance!

KarishmaB · Answer

Think about it - when you know the slope of a line, you basically know the angle it makes with the x axis. So knowing m tells us the slope of both the lines i.e. the angle made by each with the x axis and that will stay the same. It doesn't matter where they intersect. The angle made between them will just be the difference between the angles they make on the x axis. Say line 1 makes an angle of 80 degrees with the x axis and line 2 makes an angle of 50 degrees with the x axis. The angle between them will be 30 degrees. @ankhere has done a great job of showing this with a diagram here: https://gmatclub.com/forum/m08-183786.html#p1533190

GDT · Answer

VeritasKarishma

Though this is a DS ques, but still if we were to find the value of angle a, will the following process be correct?

So, if m=1 then tan θ=1; θ=45 degree and  for m=2 θ=90 degree , Is this correct? because tan 45 is 1 but tan 90 is undefined so how do we get angle for that

Thanks in advance!

KarishmaB · Answer

When m = 1, tan θ = 1, θ = 45 degrees When m = 2, tan θ = 2, θ = somewhere near 75-80 degrees (from the graph) b0314c458ecaf2909d5e63fb0e619ed2.png When θ = 90 degrees, tan θ is not defined.

dave13 · Answer

KarishmaB checked other posts but still have questions... how do we know that slope of y=x−bm is 1 ? i equated to equations of two lines (since one of the points is the same for both lines) so with statement one i got this 2x+b=-b2+x ----> so i get x=-b no b is Y-intercept and x is x-intercept ... and now what

so how can it be sufficient ?

KarishmaB · Answer

The form of the equation of a line is 
y = mx + c
Whatever is the co-efficient of x, it is the slope and c is the y intercept.

y = x - bm
is same as y = 1*x - bm
Here, co-effiicient of x is 1 so slope is 1 and -bm is the y intercept.

We solve the equations together when we want to find their point of intersection (the x and y co-ordinates of the point) but we don't care about that here. We just need the angle between the lines. The angle of a line with the X axis is completely defined by its slope. If we know the slope, we know the angle the line makes with the X axis. 
Since statement 1 gives us the slope of both the lines, we know the angles made by each line with the X axis. Say if one line makes an angle of 75 degrees and the other makes an angle of 45 degrees, we know that the angle between them is 30 degrees. 
So angle made by each line with the X axis is sufficient to give us the angle between the two lines. 
That is why statement 1 is sufficient.

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

M08-15

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