Bunuel wrote:
A
person walked completely around the edge of a park beginning at the
midpoint of one edge and
making the minimum number of turns, each with the minimum number of degrees necessary, as shown in the figure above. What is the sum of the degrees of all the turns that the person made?
(1) One of the turns is 80 degrees.
(2) The
number of sides of the park is 4, all of the sides are straight, and each interior angle is
less than 180 degrees.
DS33602.01
Quantitative Review 2020 NEW QUESTION
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Though there are couple of good posts elaborating the the solution, here's how i did it. Question keywords are highlighted here's what they mean:
1. person walked completely around the edge of a park - one lap is made by the person.
2. midpoint of one edge - I guess this is given for clarity. Had it been any of the corners of the park it would have been difficult to know the direction of the person.
3. making the minimum number of turns - Not sure about this one but i think this is reiterating the meaning of the first one. Experts please clarify...!!
Now
St. 1 is INSUFFICIENT as we all know.
St. 2 The number of sides is 4 so it is quadrilateral - does not matter if it is a rectangle(irregular)/square(regular) or anything else. As the interior angle is less than 180°, the quadrilateral is convex and this is where people normally got stuck.
Primer: Refer the easier question in the link to better understand the situation here.
https://gmatclub.com/forum/the-figure-s ... 08268.htmlNow the sum of the degrees of all the turns are given by the sum of exterior angles(nicely shown in the diagram) made at each of the points/corner.
Let the interior angles be a, b, c and d. Also, sum of interior angles of a convex quadrilateral is (n-2)*180° = 360°. So, a + b + c + d = 360°.
The exterior angle = 180° - interior angle
The four exterior angles are 180° - a, 180° - b, 180° - c and 180° - d.
Hence, Sum of exterior angles is
= 180° - a + 180° - b + 180° - c + 180° - d
= 180° * 4 - (a + b+ c + d)
= 720° - 360°
= 360°
DavidTutorexamPAL wrote:
The sum of angles the person walked is not the sum of exterior angles (which complete the interior angles to 180 as you write), but rather 360 - each internal angle (as the person makes a 'full turn' around each corner).
So the calculation you would need to to do is (360 - internal angle 1) + (360 - internal angle 2) + .... for all 4 angles.
Since the internal angles of a quadrilateral sum to 360, this gives 360*4 - 360 = 1080.
DavidTutorexamPALHi, I think you meant 180° in "(360 - internal angle 1) + (360 - internal angle 2) + .... for all 4 angles". If not, then i have to disagree with you because your solution made me think that the person was moving forward but was facing backward, finally turning(rotating) by 360° - int. angle. After the first turn s/he, facing forward, was moving forward but we would be in trouble at the second turn for how to calculate the angle. I hope i have clearly communicated my point of view.
Though this hardly matters as this is DS question, it would have been wrong had this been PS question.
Although we can calculate the solution, we need not to do any math as we are given the fixed number of sides with their respective interior angles.
Finally, if one can imagine a real world situation, it can be easily understood that
If one travels along a rectangular city block, completing one lap, s/he makes a total of 360° sum.
OR
If someone takes 1 u-turn s/he makes a 180° sum and if 2 u-turns, reaching the same position where s/he started(completing 1 lap), then 360° sum.
OR
Had the park been circular(infinite turns) in shape then the person would again have made 360° sum.
Misunderstandings people may make:
- Whether number of turns mean number of corner at which turns were taken.
- What number of complete laps around the park made? This is because of confusion one may arrive at after reading "minimum number of turns".
- An unlikely one that turns not in multiples of 4 are made i.e. person turned 6 times or 7 times. Then it would have been difficult to calculate.
Hope this is helpful..!!
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