The figure above shows two lines intersecting at the point O. If the l

D.

Qtn asks about angle of each line when both lines are moving at same rate in opposite direction.

First the angle "theta" will become zero and both lines will move by (theta/2). Then they will move till they become perpendicular to each other. Each line will move by (90/2).

Total: (90+theta)/2

hardest part was stepping back and thinking logically about the question.....that took at least a minute to a minute and half.


The Lines are moving at the Same Rate. Therefore, Assume 1 Line is moving to make it easier. Then the Distance that 1 Line has to move for the Lines to be 90 degrees is Cut in 1/2 because the Other Line will make up the other 1/2 Distance.

(1st) Since the Lines are moving in opposite directions, the 1 Line will move towards the 2nd Line until they are Exactly on top of each other. The Distance required to get to that moment is Theta Degrees (the given degrees)

(2nd) Now that they are on top of each other, the 1 Line will have to move another 90 Degrees until the 2 Lines are Perpendicular to each other.

Total movement that 1 Line would have to make = (Theta) + 90 degrees


Again, since both Lines are moving at the Same Rate in Opposite Directions, the 2nd Line will take care of 1/2 of this Distance.

So the total distance that EACH Line will move:


[(Theta) + 90] * (1/2)

-D-

We have two lines intersecting at point O.

\(θ + θ + (180-θ) + (180-θ) = 360 degrees\)

Both lines are rotating at the same rate and in the opposite direction.

Therefore each line will have rotated \(\frac{θ}{2}\) degrees when they overlap each other to form a straight line.

To go from a straight line to two perpendicular lines, each line needs to rotate 90/2 degrees.

Each line must move: \(\frac{θ + 90}{2}\)

Answer is D.

BrentGMATPrepNow if theta = 30, then option A is correct too, isnt it?

If theta = 30, then the answer to the question (the output) is 60.

In that case, answer choices A and D we'll both work, which means we need to select another value for theta and then test answer choices A and D in the hopes of eliminating one of them.
This is one of the major drawbacks of the input-output approach; the strategy doesn't always eliminate four of the five answer choices (which means more work for you)

GMATNinja, can you please the best approach to solve this question under time constraint?

Bunuel, please advice if there are variations of this question?

While solving any GMAT Quant question, the very first thing required is the CORRECT understanding of everything that is given in the question as well as that we need to find. In the same spirit, let’s begin by stating and completely understanding the Given.


GIVEN:

• Two lines AB and CD intersect at point O, with an angle of θ° between them.
• Both lines are rotated about O at the same rate and in opposite directions until AB⊥CD.

Note: The overall movement of the lines can be segmented into two parts:

- Part 1: This is when the lines move towards each other (until they overlap), and ---- (I)
- Part 2: This is where the lines, after overlapping in (I), move away from each other until AB becomes perpendicular to CD. ---- (II)

TO FIND:

• The degrees by which each line moved overall.

APPROACH:
Since it is a geometry question, the process skill of visualization will make everything super clear. So, we will visualize each step and then convert our visualizations into mathematical expressions.


WORKING:
The initial positions of the lines AB and CD (as given) are shown below:


Part 1 of the movement:
From (I), we know that first AB and CD move towards each other, until they overlap.
Observe below that after covering a combined TOTAL of θ° (because currently they are θ° apart), AB and CD will coincide, as shown in the below image.


So, till now, the COMBINED angle moved by AB and CD = θ°. ----(III)

Now, since both the lines are moving at the same rate (given), the angle moved by each of the them individually should be the same. Hence, using (III), we can say that:

- Angle moved by EACH line so far = (θ/2)° ---- (IV)

Part 2 of the movement:
Now, after the overlap, the two lines will continue to move in their respective directions, until they become perpendicular to each other, that is, the angle between AB and CD becomes 90°.


So, after resuming their movement from the overlap point, till the time AB and CD become perpendicular to each other, the total angle moved by the two lines combined = 90°. (That is, movement from 0° to 90°).
Using (II) again, we can say that:

- Angle moved by the EACH line in this part of the movement = (90/2)° ---- (V)

Required #degrees moved by each line:
From (IV) and (V), we get:

- The total number of degrees that EACH line moved through = (θ/2)° + (90/2)°

= \(\frac{(θ + 90)}{2}\)

Correct Answer: Choice D


TAKEAWAYS:
Visualization is a key skill for almost all Geometry questions. If you can visualize a given problem correctly, you will always find yourself more comfortable and confident in solving it further.


Best,
Aditi Gupta
Quant expert, e-GMAT