Last visit was: 16 Jul 2024, 13:36 It is currently 16 Jul 2024, 13:36
Close
GMAT Club Daily Prep
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.
Close
Request Expert Reply
Confirm Cancel
SORT BY:
Date
Tags:
Show Tags
Hide Tags
Retired Moderator
Joined: 19 Oct 2018
Posts: 1868
Own Kudos [?]: 6646 [11]
Given Kudos: 705
Location: India
Send PM
Senior Moderator - Masters Forum
Joined: 19 Jan 2020
Posts: 3128
Own Kudos [?]: 2811 [1]
Given Kudos: 1511
Location: India
GPA: 4
WE:Analyst (Internet and New Media)
Send PM
RC & DI Moderator
Joined: 02 Aug 2009
Status:Math and DI Expert
Posts: 11472
Own Kudos [?]: 34380 [3]
Given Kudos: 322
Send PM
Senior Moderator - Masters Forum
Joined: 19 Jan 2020
Posts: 3128
Own Kudos [?]: 2811 [1]
Given Kudos: 1511
Location: India
GPA: 4
WE:Analyst (Internet and New Media)
Send PM
Re: There is a point P inside a regular hexagon such that area of triangle [#permalink]
1
Bookmarks
chetan2u wrote:
yashikaaggarwal wrote:
Bunuel chetan2u Kindly solve this question.



An immediate answer.
The opposite sides are parallel and distance between each set of parallel sides will be equal.

The height from P to each opposite parallel side would add up to H.

Note- On mobile, so will add a detailed explanation and sketch.

But the hexagon can be divided into equal areas by adding the areas of alternate triangles.
PAB+PCD+PEF=PBC+PDE+PFA=8+5+3=16
So total area =16*2=32.

Now the opposite triangles should add up to 1/3 of total area as there are 3 sets of opposite triangle whose heights add up to same H and also the base is same.
So each set of opposite triangles will have an area 32/3.
So area of highlighted triangle = 32/3-8=8/3


B

I didn't get the green bold highlight, can you explain it with diagram? Thank you
Manager
Manager
Joined: 05 Jan 2020
Posts: 145
Own Kudos [?]: 134 [5]
Given Kudos: 288
Send PM
There is a point P inside a regular hexagon such that area of triangle [#permalink]
3
Kudos
1
Bookmarks
h1 = 16/s, h2 = 10/s, h3 = 6/s
Height of equilateral triangle PQR = h1+h2+h3 = 32/s
Height of hexagon ABCDEF = (2/3)*Height of triangle PQR = 64/3s
=> h1 + h = 64/3s
=> h = 64/s - h1 = 64/3s - 16/s
=> h = 16/3s

Area = 1/2 * s * 16/3s = 8/3

Edit: Point inside hexagon is denoted by O.
Attachments

Untitled.png
Untitled.png [ 8.65 KiB | Viewed 2572 times ]


Originally posted by Lipun on 15 Aug 2020, 21:50.
Last edited by Lipun on 16 Aug 2020, 10:17, edited 1 time in total.
Retired Moderator
Joined: 19 Oct 2018
Posts: 1868
Own Kudos [?]: 6646 [0]
Given Kudos: 705
Location: India
Send PM
Re: There is a point P inside a regular hexagon such that area of triangle [#permalink]
Brilliant Bhai?

Lipun wrote:
h1 = 16/s, h2 = 10/s, h3 = 6/s
Height of equilateral triangle PQR = h1+h2+h3 = 32/s
Height of hexagon ABCDEF = (2/3)*Height of triangle PQR = 64/3s
=> h1 + h = 64/3s
=> h = 64/s - h1 = 64/3s - 16/s
=> h = 16/3s

Area = 1/2 * s * 16/3s = 8/3


Posted from my mobile device
SVP
SVP
Joined: 27 May 2012
Posts: 1696
Own Kudos [?]: 1493 [0]
Given Kudos: 639
Send PM
Re: There is a point P inside a regular hexagon such that area of triangle [#permalink]
Lipun wrote:
h1 = 16/s, h2 = 10/s, h3 = 6/s
Height of equilateral triangle PQR = h1+h2+h3 = 32/s
Height of hexagon ABCDEF = (2/3)*Height of triangle PQR = 64/3s
=> h1 + h = 64/3s
=> h = 64/s - h1 = 64/3s - 16/s
=> h = 16/3s

Area = 1/2 * s * 16/3s = 8/3


Hi Lipun,
Can you please elaborate what relation you used to find the height of H1 , H2 and H3?
Manager
Manager
Joined: 05 Jan 2020
Posts: 145
Own Kudos [?]: 134 [1]
Given Kudos: 288
Send PM
Re: There is a point P inside a regular hexagon such that area of triangle [#permalink]
1
Kudos
stne wrote:

Hi Lipun,
Can you please elaborate what relation you used to find the height of H1 , H2 and H3?


Area of triangle PDE = 1/2*h2*s = 5 => h2 = 10/s. Similarly, for h1 and h3.
The height of an equilateral triangle is equal to the sum of the perpendicular distances of a point P from each of the sides.
Thus, H = h1+h2+h3 = 32/s

Hope this answers your query!
SVP
SVP
Joined: 27 May 2012
Posts: 1696
Own Kudos [?]: 1493 [0]
Given Kudos: 639
Send PM
Re: There is a point P inside a regular hexagon such that area of triangle [#permalink]
Lipun wrote:
stne wrote:

Hi Lipun,
Can you please elaborate what relation you used to find the height of H1 , H2 and H3?


Area of triangle PDE = 1/2*h2*s = 5 => h2 = 10/s. Similarly, for h1 and h3.
The height of an equilateral triangle is equal to the sum of the perpendicular distances of a point P from each of the sides.
Thus, H = h1+h2+h3 = 32/s

Hope this answers your query!


Hi Lipun ,
Can you please check your figure, there are 2 points marked as P, causing some confusion in understanding your first explanation.
Thank you.
Manager
Manager
Joined: 05 Jan 2020
Posts: 145
Own Kudos [?]: 134 [0]
Given Kudos: 288
Send PM
Re: There is a point P inside a regular hexagon such that area of triangle [#permalink]
stne wrote:

Hi Lipun ,
Can you please check your figure, there are 2 points marked as P, causing some confusion in understanding your first explanation.
Thank you.


Thanks stne for pointing out! Have edited to denote it as O.
Area of ODE = 1/2*h2*s = 5 => h2 = 10/s

Lipun
Intern
Intern
Joined: 18 Jun 2023
Posts: 3
Own Kudos [?]: 0 [0]
Given Kudos: 74
Send PM
Re: There is a point P inside a regular hexagon such that area of triangle [#permalink]
I still don't understand after going through the responses here, can someone please give another explanation?
GMAT Club Bot
Re: There is a point P inside a regular hexagon such that area of triangle [#permalink]
Moderator:
Math Expert
94370 posts