NEW HERE? LEARN MORE
closeclose
Last visit was: 26 Apr 2024, 03:45 It is currently 26 Apr 2024, 03:45
Decision Tracker
My Rewards
New posts
New comers' posts

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.
Go to My Error Log Learn more
Close
Request Expert Reply
Confirm Cancel

The lengths of the two shorter legs of a right triangle add up to 40

SORT BY:
Date
Kudos
Tags:
Find Similar Topics 
Show Tags
Hide Tags
User avatar
Lstadt
Manager
Manager
Joined: 08 Jun 2011
Posts: 62
Own Kudos [?]: 76 [15]
Given Kudos: 65
Send PM
The lengths of the two shorter legs of a right triangle add up to 40 [#permalink] New post   Updated on: 10 Mar 2021, 13:39
15
Bookmarks
00:00
A
B
C
D
E

Difficulty:

15% (low)

Question Stats:

76% (01:12) correct 24% (01:20) wrong based on 216 sessions
Hide Show timer Statistics
The lengths of the two shorter legs of a right triangle add up to 40 units. What is the maximum possible area of the triangle?

A. 40 square units

B. 80 square units

C. 120 square units

D. 200 square units

E. 400 square units


MGMAT Solution:
Show Spoiler
200 Square Units. You can think of a triangle as half of a rectangle. Constructing this right triangle with legs adding up to 40 is equivalent to constructing the rectangle with a perimeter of 80. Since the area of the triangle is half that of the rectangle, you can use the previously mentioned technique for maximizing the area of the rectangle: of all rectangles with a given perimeter, the square has the greatest area. The desired rectangle is thus 20 by 20 square and the right triangle has the area (1/2)(20)(20)=200 units.

Answer is 200


Question:
Show Spoiler
MGMT set it up as a square then went from there by calculating it as half the area of a maximized square.

My approach was different and it yielded a wrong result, of course :), but I am sure that I had the right idea in mind and something went wrong.

Since we want to maximize a triangle, the maximum area would be that of an Equalitral triangle which has 3 equal sides. Since s1+s2 = 40 (assuming we are dealing with an Equalitral) then S1 must be 20; therefore, s2 and s3 has to be each 20.

The area of an Equalitral triangle is s^2\sqrt{3}/4 which should give us
400 * \sqrt{3}/4 which is 100\sqrt{3}.

What did I do wrong there?


OA:
Show Spoiler
200
ShowHide Answer
Official Answer
D

Originally posted by Lstadt on 28 Feb 2012, 13:48.
Last edited by Bunuel on 10 Mar 2021, 13:39, edited 3 times in total.
Renamed the topic and edited the question.
Most Helpful Reply
User avatar
L
Bunuel
Math Expert
Joined: 02 Sep 2009
Posts: 92929
Own Kudos [?]: 619112 [11]
Given Kudos: 81609
Send PM
CAT Tests Forum Quiz Mode
Re: The lengths of the two shorter legs of a right triangle add up to 40 [#permalink] New post   28 Feb 2012, 14:02
4
Kudos
7
Bookmarks
Expert Reply
Lstadt wrote:
Question is from MGMT

The lengths of the two shorter legs of a right triangle add up to 40 units. What is the maximum possible area of the triangle?

Answer is
Show Spoiler
200


MGMT set it up as a square then went from there by calculating it as half the area of a maximized square.

My approach was different and it yielded a wrong result, of course :), but I am sure that I had the right idea in mind and something went wrong.

Since we want to maximize a triangle, the maximum area would be that of an Equalitral triangle which has 3 equal sides. Since s1+s2 = 40 (assuming we are dealing with an Equalitral) then S1 must be 20; therefore, s2 and s3 has to be each 20.

The area of an Equalitral triangle is s^2\sqrt{3}/4 which should give us
400 * \sqrt{3}/4 which is 100\sqrt{3}.

What did I do wrong there?


The lengths of the two shorter legs of a right triangle add up to 40 units. What is the maximum possible area of the triangle?

First of all: a right triangle can never be equilateral, since the hypotenuse (the side opposite the right angle) is always longer than any the other two sides.

Next, a right triangle with the largest area will be an isosceles right triangle, so the the lengths of the two shorter legs must be equal, so each must be 20. Area=1/2*20*20=200.

Hope it's clear.
User avatar
L
Bunuel
Math Expert
Joined: 02 Sep 2009
Posts: 92929
Own Kudos [?]: 619112 [6]
Given Kudos: 81609
Send PM
CAT Tests Forum Quiz Mode
Re: The lengths of the two shorter legs of a right triangle add up to 40 [#permalink] New post   29 Feb 2012, 10:31
1
Kudos
5
Bookmarks
Expert Reply
pbull78 wrote:
Nice expanatio bunuel , bunuel can u tell me link from where I can get all these formulaes of geometry section
Thanks

Posted from my mobile device


Check geometry chapters of Math Book:
TRIANGLES: https://gmatclub.com/forum/math-triangles-87197.html
POLYGONS: https://gmatclub.com/forum/math-polygons-87336.html
COORDINATE GEOMETRY: https://gmatclub.com/forum/math-coordina ... 87652.html
CIRCLES: https://gmatclub.com/forum/math-circles-87957.html
3-D Geometries: https://gmatclub.com/forum/math-3-d-geom ... ml#p792331

Hope it helps.
General Discussion
User avatar
B
AbeinOhio
Manager
Manager
Joined: 22 Feb 2012
Posts: 74
Own Kudos [?]: 67 [0]
Given Kudos: 25
Schools: HBS '16
GMAT 1: 670 Q42 V40
GMAT 2: 740 Q49 V42
GPA: 3.47
WE:Corporate Finance (Aerospace and Defense)
Send PM
GMAT ToolKit User
Re: The lengths of the two shorter legs of a right triangle add up to 40 [#permalink] New post   28 Feb 2012, 14:14
I did this in a slightly different and clearly more difficult way than presented above but maybe it will help someone else..

I set the 2 shorter sides to be x & y

so:
x + y = 40

&

then (X+Y)^2 = 1600

The area of this triangle= xy/2

I expanded the (X+Y)^2 expression and moved XY to one side to get:
xy= (1600 -X^2 - y^2)/2

so XY/2 = (1600-X^2- y^2)4

Now to maximize this expression i tried extremes (39,1) (38,2) and realized that it was maxmized when X= Y

and was able to solve so that (1600-20^2-20^2)/4 = 200

Hope that helps you guys.
User avatar
L
Bunuel
Math Expert
Joined: 02 Sep 2009
Posts: 92929
Own Kudos [?]: 619112 [2]
Given Kudos: 81609
Send PM
CAT Tests Forum Quiz Mode
Re: The lengths of the two shorter legs of a right triangle add up to 40 [#permalink] New post   28 Feb 2012, 15:05
1
Kudos
1
Bookmarks
Expert Reply
AbeinOhio wrote:
I did this in a slightly different and clearly more difficult way than presented above but maybe it will help someone else..

I set the 2 shorter sides to be x & y

so:
x + y = 40

&

then (X+Y)^2 = 1600

The area of this triangle= xy/2

I expanded the (X+Y)^2 expression and moved XY to one side to get:
xy= (1600 -X^2 - y^2)/2

so XY/2 = (1600-X^2- y^2)4

Now to maximize this expression i tried extremes (39,1) (38,2) and realized that it was maxmized when X= Y

and was able to solve so that (1600-20^2-20^2)/4 = 200

Hope that helps you guys.


Consider this, we are given that a+b=40 and we want to maximize area=1/2*ab, so basically we want to maximize the value of ab. Now, for a given sum of two values the product is maximized when they are equal. Hence ab is maximized when a=b=20.

Basically the property saying that a right triangle with the largest area will be an isosceles right triangle is derived from that rule.

Hope it helps.
avatar
pbull78
Intern
Intern
Joined: 16 Dec 2011
Posts: 28
Own Kudos [?]: 19 [0]
Given Kudos: 12
GMAT Date: 04-23-2012
Send PM
Re: The lengths of the two shorter legs of a right triangle add up to 40 [#permalink] New post   29 Feb 2012, 09:03
Nice expanatio bunuel , bunuel can u tell me link from where I can get all these formulaes of geometry section
Thanks

Posted from my mobile device
avatar
PareshGmat
SVP
SVP
Joined: 27 Dec 2012
Status:The Best Or Nothing
Posts: 1562
Own Kudos [?]: 7208 [0]
Given Kudos: 193
Location: India
Concentration: General Management, Technology
WE:Information Technology (Computer Software)
Send PM
Re: The lengths of the two shorter legs of a right triangle add up to 40 [#permalink] New post   30 Sep 2014, 01:44
Given that sum of two shorter sides of right triangle = 40

It means, they can be in any combination from 1 - 39, 2 - 38, 3-37.............. upto 39 - 1

The mean of this would be 20 - 20 (Means its a Isosceles right triangle)

When sides are 20 - 20, area would be maximum \(= \frac{1}{2} * 20 * 20 = 200\)

Please note: If combinations like 21 - 19, 22 - 18, 23 - 17 etc. are tried, there product would always be less than 400
User avatar
bumpbot
Non-Human User
Joined: 09 Sep 2013
Posts: 32681
Own Kudos [?]: 822 [0]
Given Kudos: 0
Send PM
Re: The lengths of the two shorter legs of a right triangle add up to 40 [#permalink] New post   19 Nov 2023, 07:13
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
GMAT Club Bot
gmatclubot
Re: The lengths of the two shorter legs of a right triangle add up to 40 [#permalink]
19 Nov 2023, 07:13
Moderators:
Bunuel
Math Expert
92929 posts
yashikaaggarwal
Senior Moderator - Masters Forum
3137 posts

Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne
Main navigation
GMAT Resources
Partners
MBA Resources
www.gmac.com|www.mba.com | | GMAT Club Rules | Terms and Conditions