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605-655 Level|   Geometry|               
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KarishmaB
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A person walked completely around the edge of a park beginning at the 29 Jun 2020, 03:24
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GDT
VeritasKarishma
Bunuel

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


Show Spoiler
Attachment:
2019-04-26_1836.png


Responding to a pm:

You need to find the sum of exterior angles of a polygon. We know that the sum of exterior angles of a convex polygon is always 360. But the question stem does not give us that the polygon is convex. What if the park looks like this:

Attachment:
Picture_3.png
Certainly feasible! But this is a concave polygon and sum of all angles turned here will not be 360.
So what we need from the statements is whether the park is a convex polygon or not.

(1) One of the turns is 80 degrees.
No information about other angles.

(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.
"Each interior angle is < 180" means it is a convex polygon. Sufficient.


Answer (B)


VeritasKarishma

I have a doubt in the highlighted portion

If the park is a convex polygon, say a regular pentagon, wouldn't the turns be taken around vertex and so the degree of turn should be interior angle, and in case it is not convex as shown in the image attached by you, then for the obtuse angle we'll take exterior angle's measure. That's why we need to know if it is convex + the no. of sides

Kindly clarify!
Show more

This link shows you what exterior angles are. They are the same angles that are turned by the person.
https://www.mathsisfun.com/geometry/ext ... ygons.html

They are NOT the complete angles around the vertex.
Sum of exterior angles of a convex polygon are always 360. Think why.
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A person walked completely around the edge of a park beginning at the 29 Jun 2020, 04:13
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VeritasKarishma
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VeritasKarishma
Bunuel

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


Show Spoiler
Attachment:
2019-04-26_1836.png


Responding to a pm:

You need to find the sum of exterior angles of a polygon. We know that the sum of exterior angles of a convex polygon is always 360. But the question stem does not give us that the polygon is convex. What if the park looks like this:

Attachment:
Picture_3.png
Certainly feasible! But this is a concave polygon and sum of all angles turned here will not be 360.
So what we need from the statements is whether the park is a convex polygon or not.

(1) One of the turns is 80 degrees.
No information about other angles.

(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.
"Each interior angle is < 180" means it is a convex polygon. Sufficient.


Answer (B)


VeritasKarishma

I have a doubt in the highlighted portion

If the park is a convex polygon, say a regular pentagon, wouldn't the turns be taken around vertex and so the degree of turn should be interior angle, and in case it is not convex as shown in the image attached by you, then for the obtuse angle we'll take exterior angle's measure. That's why we need to know if it is convex + the no. of sides

Kindly clarify!

This link shows you what exterior angles are. They are the same angles that are turned by the person.
https://www.mathsisfun.com/geometry/ext ... ygons.html

They are NOT the complete angles around the vertex.
Sum of exterior angles of a convex polygon are always 360. Think why.
Show more

VeritasKarishma

I understood why exterior angles will always add up to 360 because interior angles sum= (n-2)*180 degrees

My ques was why we are taking sum of exterior angles instead of interior angles. When a person makes a turn, the degrees he has turned should be the measure of interior angle Isn't it?

I don't know how to attach a picture in post, but hope you'll get what my doubt is

Thanks in advance!
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A person walked completely around the edge of a park beginning at the 01 Jul 2020, 00:38
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VeritasKarishma
Bunuel

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


Show Spoiler
Attachment:
2019-04-26_1836.png


Responding to a pm:

You need to find the sum of exterior angles of a polygon. We know that the sum of exterior angles of a convex polygon is always 360. But the question stem does not give us that the polygon is convex. What if the park looks like this:

Attachment:
Picture_3.png
Certainly feasible! But this is a concave polygon and sum of all angles turned here will not be 360.
So what we need from the statements is whether the park is a convex polygon or not.

(1) One of the turns is 80 degrees.
No information about other angles.

(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.
"Each interior angle is < 180" means it is a convex polygon. Sufficient.


Answer (B)


VeritasKarishma

I have a doubt in the highlighted portion

If the park is a convex polygon, say a regular pentagon, wouldn't the turns be taken around vertex and so the degree of turn should be interior angle, and in case it is not convex as shown in the image attached by you, then for the obtuse angle we'll take exterior angle's measure. That's why we need to know if it is convex + the no. of sides

Kindly clarify!

This link shows you what exterior angles are. They are the same angles that are turned by the person.
https://www.mathsisfun.com/geometry/ext ... ygons.html

They are NOT the complete angles around the vertex.
Sum of exterior angles of a convex polygon are always 360. Think why.

VeritasKarishma

I understood why exterior angles will always add up to 360 because interior angles sum= (n-2)*180 degrees

My ques was why we are taking sum of exterior angles instead of interior angles. When a person makes a turn, the degrees he has turned should be the measure of interior angle Isn't it?

I don't know how to attach a picture in post, but hope you'll get what my doubt is

Thanks in advance!
Show more

GDT, when a person turns, the angle turned is the exterior angle, not the interior angle.
Look at the original diagram here: https://gmatclub.com/forum/a-person-wal ... l#p2266301

See the man walking... At some point, he decides to turn. He would have otherwise continued to go straight. He turns and the angle shown by the curved arrow is the angle he turns. Then he starts going straight again and after some time, instead of going straight, he turns by the angle shown by the other curved arrow. This if he does another time, we get a quadrilateral. The angles shown by curved arrows are exterior angles of the quadrilateral.
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A person walked completely around the edge of a park beginning at the 07 Oct 2020, 01:10
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Bunuel

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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Attachment:
2019-04-26_1836.png
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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.html

Now 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
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.
Show more
DavidTutorexamPAL
Hi, 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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A person walked completely around the edge of a park beginning at the 04 Nov 2020, 18:32
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Bunuel

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
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(1) one of the turns is 80 degrees. this may not be a problem so much except we dont know how many sides there even are to this park, thus we dont know how many turns which are external angles of our polygon, or partial polygon NS

(2) we are told the park has 4 sides, all straight and each interior angle is less than 180 degrees. While this would seem NS since we dont have any individual angle measure, with geometry there is always that chance that variables can drop out in our calculation, so go ahead and try to set up a drawing. It needs to be four sides. also extend the sides so we have external angles on all sides. Label the internal angles x,y,z and the last one will be (360 - x - y -z) since any four sided object internal angles sum to 360. Now external angles form straight angles with their paired interior angle, so each exterior angle will be 180 minus our value. So our exterior angles are 180-x, 180-y, 180-z, and 180-(360-x-y-z). Sum these up and we get sum = 180 -x + 180 -y + 180-z +180 - 360 +x+y+z. so our variables drop out and we get 3(180) - 2(180) = 180 sufficient OA is B

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A person walked completely around the edge of a park beginning at the 17 Dec 2020, 12:21
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Bunuel

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.
Show more

(1) Other turns may be 190 Degrees or less than that. Insufficient.

(2) So, it is a quadrilateral. The turns will be equal to the sum of the interior angles. \(180(4-2);\) Sufficient.

The answer is \(B\)
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A person walked completely around the edge of a park beginning at the 04 Dec 2022, 13:54
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KarishmaB
Bunuel

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


Show Spoiler
Attachment:
2019-04-26_1836.png


Responding to a pm:

You need to find the sum of exterior angles of a polygon. We know that the sum of exterior angles of a convex polygon is always 360. But the question stem does not give us that the polygon is convex. What if the park looks like this:

Attachment:
Picture_3.png
Certainly feasible! But this is a concave polygon and sum of all angles turned here will not be 360.
So what we need from the statements is whether the park is a convex polygon or not.

(1) One of the turns is 80 degrees.
No information about other angles.

(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.
"Each interior angle is < 180" means it is a convex polygon. Sufficient.


Answer (B)
Show more

KarishmaB
Thank you for your helpful reply. To clarify, for both concave and convex polygons, the sum of the exterior angles are always 360?

Also "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" just is a fancy of way of asking for the exterior angles, correct? Tricky!
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A person walked completely around the edge of a park beginning at the 04 Dec 2022, 22:24
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KarishmaB
Bunuel

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


Show Spoiler
Attachment:
2019-04-26_1836.png


Responding to a pm:

You need to find the sum of exterior angles of a polygon. We know that the sum of exterior angles of a convex polygon is always 360. But the question stem does not give us that the polygon is convex. What if the park looks like this:

Attachment:
Picture_3.png
Certainly feasible! But this is a concave polygon and sum of all angles turned here will not be 360.
So what we need from the statements is whether the park is a convex polygon or not.

(1) One of the turns is 80 degrees.
No information about other angles.

(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.
"Each interior angle is < 180" means it is a convex polygon. Sufficient.


Answer (B)

KarishmaB
Thank you for your helpful reply. To clarify, for both concave and convex polygons, the sum of the exterior angles are always 360?

Also "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" just is a fancy of way of asking for the exterior angles, correct? Tricky!
Show more

In case of convex polygons, exterior angles are the angles turned at every change in direction. For convex polygons, the sum of all exterior angles is simply 360 degrees.

When it comes to concave polygons, I am a little unsure about acceptance of the same terminology. Exterior angles are still the angles turned but note that one angle is turned in the opposite direction making it negative. So we won't add it up but instead subtract it and overall, then, the sum of all the exterior angles will still be 360 degrees. To start from a point and turn and come back to the same point, one would need to travel overall 360 degrees.
But if I were to just add the degrees turned (not worry about the direction in which we turned), then the overall sum of degrees turned would be more than 360.

This question talks about the sum of the degrees turned (no subtraction) so that would be 360 if the polygon is a convex polygon. Since statement 2 tells us that the polygon is convex (all interior angles are less than 180 degrees), we know that the sum of degrees turned will be 360.
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A person walked completely around the edge of a park beginning at the 08 Jan 2025, 16:51
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