Bunuel wrote:
The figure above shows two lines intersecting at the point O. If the lines are rotated about O at the same rate and in the directions shown until \(AB \bot CD\), through how many degrees must each line move?
A. \(90 - \theta\)
B. \(90 - \frac{\theta}{2}\)
C. \(90 + \frac{\theta}{2}\)
D. \(\frac{90 + \theta}{2}\)
E. \(\frac{90 + 2\theta}{2}\)
PS20429
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)°
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 _________________