BREAKING: Target Test Prep releases Brand New 2026 On Demand GMAT prep course

Redeem

The hypotenuse of a right triangle is 10 cm...

This topic has expert replies
Post new topic Post Reply
Previous Topic Next Topic
Junior | Next Rank: 30 Posts
Posts: 29
Joined: Wed Nov 20, 2013 5:35 pm
Thanked: 1 times

The hypotenuse of a right triangle is 10 cm...

by Zach.J.Dragone » Fri Dec 06, 2013 12:09 pm
The hypotenuse of a right triangle is 10 cm. What is the perimeter, in centimeters, of the triangle?

(1) The area of the triangle is 25 square centimeters.
(2) The 2 legs of the triangle are of equal length.

D

(1) The area of the triangle is 25 square centimeters.

So, we know that the triangle is a right triangle and that it's area is 25. We need to find x+y+10 = P

We know that A = 25 so 25 = 1/2 (b*h) --> 50=bh
Also, because this is a right triangle, we know that a^2 + b^2 = c^2 so a^2 + b^2 = 10^2. This is where I am lost.

One of the solutions I have seen is this:

Square x+y --> (x+y)^2=x^2+2xy+y^2=(x^2+y^2)+2xy=100+2*50=200 --> x+y=√200.

Thus P=x+y+10=√200+10.

Why do we square x+y? I see that they plugged in 100 for (x^2 + y^2) and 50 for 2xy, but why do they square x+y?





[/spoiler]
Top
Source: — Problem Solving |

GMAT/MBA Expert

User avatar
GMAT Instructor
Posts: 768
Joined: Wed Dec 28, 2011 4:18 pm
Location: Berkeley, CA
Thanked: 387 times
Followed by:140 members

by Mike@Magoosh » Fri Dec 06, 2013 1:26 pm
Zach.J.Dragone wrote:The hypotenuse of a right triangle is 10 cm. What is the perimeter, in centimeters, of the triangle?

(1) The area of the triangle is 25 square centimeters.
(2) The 2 legs of the triangle are of equal length.

D

(1) The area of the triangle is 25 square centimeters.

So, we know that the triangle is a right triangle and that it's area is 25. We need to find x+y+10 = P

We know that A = 25 so 25 = 1/2 (b*h) --> 50=bh
Also, because this is a right triangle, we know that a^2 + b^2 = c^2 so a^2 + b^2 = 10^2. This is where I am lost.

One of the solutions I have seen is this:

Square x+y --> (x+y)^2=x^2+2xy+y^2=(x^2+y^2)+2xy=100+2*50=200 --> x+y=√200.

Thus P=x+y+10=√200+10.

Why do we square x+y? I see that they plugged in 100 for (x^2 + y^2) and 50 for 2xy, but why do they square x+y?
Zach.J.Dragone
I'm happy to help. :-)

First of all, I am going to recommend a post I think you will find very helpful:
https://magoosh.com/gmat/2013/how-to-do- ... th-faster/
What are asking has very much to do with pattern-matching and the way it plays out in math solutions.

Here is a blog about some algebra formulas:
https://magoosh.com/gmat/2013/three-alge ... -the-gmat/
The three formulas in that blog are absolutely essential to memorize.

What the person who wrote the solution saw was
1) from Pythagorean Theorem, we have (x^2) + (y^2) = 100
2) from area, .5*xy = 25 --> xy = 50
From these, we need (x + y), no easy task. But, and here's the brilliant left-brain flash of insight --- if we square (x + y), then the pattern that results will incorporate all those pieces. This requires knowing from memory that:
(x + y)^2 = (x^2) + 2xy + (y^2)
If you don't have that formula memorized already, then there's no way you would be able to see that that pieces of the right side of that formula can be rearranged in terms of the two pieces of information we have.

Does all this make sense?
Mike :-)
Magoosh GMAT Instructor
https://gmat.magoosh.com/
Top
Junior | Next Rank: 30 Posts
Posts: 29
Joined: Wed Nov 20, 2013 5:35 pm
Thanked: 1 times

by Zach.J.Dragone » Fri Dec 06, 2013 1:43 pm
Thanks for the response Mike!

I am quite familiar with those formulas but I didn't make the connection. I understand we need them (and how the numbers are plugged in) but I guess my problem is why do we need them? How do I know that on the test? I'm not sure that's a question anyone can answer for me...even after doing truck loads of problems I still miss so many because I fail to see these tricks and clues as to what to do next. As soon as I see the explanation (oftentimes simply as soon as I see the correct answer) I know exactly what I did wrong and how to solve the problem, but for some reason I'm just not picking up on all these "signs" in the questions!
Mike@Magoosh wrote:
Zach.J.Dragone wrote:The hypotenuse of a right triangle is 10 cm. What is the perimeter, in centimeters, of the triangle?

(1) The area of the triangle is 25 square centimeters.
(2) The 2 legs of the triangle are of equal length.

D

(1) The area of the triangle is 25 square centimeters.

So, we know that the triangle is a right triangle and that it's area is 25. We need to find x+y+10 = P

We know that A = 25 so 25 = 1/2 (b*h) --> 50=bh
Also, because this is a right triangle, we know that a^2 + b^2 = c^2 so a^2 + b^2 = 10^2. This is where I am lost.

One of the solutions I have seen is this:

Square x+y --> (x+y)^2=x^2+2xy+y^2=(x^2+y^2)+2xy=100+2*50=200 --> x+y=√200.

Thus P=x+y+10=√200+10.

Why do we square x+y? I see that they plugged in 100 for (x^2 + y^2) and 50 for 2xy, but why do they square x+y?
Zach.J.Dragone
I'm happy to help. :-)

First of all, I am going to recommend a post I think you will find very helpful:
https://magoosh.com/gmat/2013/how-to-do- ... th-faster/
What are asking has very much to do with pattern-matching and the way it plays out in math solutions.

Here is a blog about some algebra formulas:
https://magoosh.com/gmat/2013/three-alge ... -the-gmat/
The three formulas in that blog are absolutely essential to memorize.

What the person who wrote the solution saw was
1) from Pythagorean Theorem, we have (x^2) + (y^2) = 100
2) from area, .5*xy = 25 --> xy = 50
From these, we need (x + y), no easy task. But, and here's the brilliant left-brain flash of insight --- if we square (x + y), then the pattern that results will incorporate all those pieces. This requires knowing from memory that:
(x + y)^2 = (x^2) + 2xy + (y^2)
If you don't have that formula memorized already, then there's no way you would be able to see that that pieces of the right side of that formula can be rearranged in terms of the two pieces of information we have.

Does all this make sense?
Mike :-)
[/i]
Top

GMAT/MBA Expert

User avatar
Elite Legendary Member
Posts: 10392
Joined: Sun Jun 23, 2013 6:38 pm
Location: Palo Alto, CA
Thanked: 2867 times
Followed by:511 members
GMAT Score:800

by [email protected] » Fri Dec 06, 2013 1:49 pm
Hi Zach,

One of the interesting things about the GMAT is that most questions can be approached in a few different ways, so it is beneficial to be a "strong thinker" on Test Day. Knowing how to do math is sometimes what's needed (although sometimes the math can be done in several different ways); in other situations, the ability to figure out patterns is what's needed.

Whoever wrote this explanation is taking a real heavy "math" approach to the problem, which is fine. I think you'll find this next approach to be a bit easier...

The prompt tells us that the hypotenuse of a right triangle is 10. The question asks us for the perimeter, so we're going to need the lengths of the other two sides.

Since we have a right triangle, we can use the Pythagorean Theorem:

X^2 + Y^2 = 10^2

**Important: triangles can't have "negative" sides, so both X and Y MUST be positive.**

We have 2 variables, but only 1 equation, so we can't figure out the exact values of X and Y

Fact 1: Triangle area = 25

Since Area = (1/2)(B)(H) we know that....

(1/2)(X)(Y) = 25
(X)(Y) = 50

Now we have ANOTHER equation. Combined with what we were given in the beginning, we have...

Two variables AND two UNIQUE equations, which means we have a "system" of equations and we CAN solve for X and Y. No more math is needed.
Fact 1 is SUFFICIENT.

Fact 2: The two legs of the triangle are equal.

Now we know that X = Y.

This gives us a SECOND equation to work with. Just as in Fact 1, we have a "system" and we can solve for X and Y.
Fact 2 is SUFFICIENT

Final Answer: D

Be on the lookout for "system" questions. You'll see a couple on Test Day and they're built around some useful math "patterns" that you can use to avoid doing work.

GMAT assassins aren't born, they're made,
Rich
Contact Rich at [email protected]
Image
Top
Junior | Next Rank: 30 Posts
Posts: 29
Joined: Wed Nov 20, 2013 5:35 pm
Thanked: 1 times

by Zach.J.Dragone » Fri Dec 06, 2013 2:05 pm
Thinking about what you said, regarding "multiple" ways to solve the problems, I was thinking that I could isolate one variable and plug into another equation as follows:

x^2+y^2=100
x+y=10
xy=50

x=10-y --> (10-y)*y=50 --> 10y-y^2 = 50 --> y^2-10y+50 = 0

But I don't think that can be factored. What am I missing here?
Top
User avatar
GMAT Instructor
Posts: 15539
Joined: Tue May 25, 2010 12:04 pm
Location: New York, NY
Thanked: 13060 times
Followed by:1906 members
GMAT Score:790

by GMATGuruNY » Fri Dec 06, 2013 2:22 pm
Zach.J.Dragone wrote:The hypotenuse of a right triangle is 10 cm. What is the perimeter, in centimeters, of the triangle?

(1) The area of the triangle is 25 square centimeters.
(2) The 2 legs of the triangle are of equal length.
Start with statement 2, which seems easier to evaluate.

Statement 2: The 2 legs of the triangle are of equal length
In an isosceles right triangle, the sides are in the following ratio: s : s : s√2.
Since the hypotenuse = s√2 = 10, s = 10/√2.
Thus, p = 10/√2 + 10/√2 + 10.
SUFFICIENT.

Statement 1: The area of the triangle is 25 square centimeters
Only ONE case satisfies statement 2 above: a triangle with sides of 10/√2, 10/√2, and 10.
Since the two statements cannot contradict each other, this case MUST also satisfy statement 1.
In other words, the area of a triangle with sides of 10/√2, 10/√2, and 10 must be 25:
(1/2)(10/√2)(10/√2) = 100/4 = 25.

Check whether this is the ONLY case that satisfies statement 1.
Another right triangle with a hypotenuse of 10 is a 6-8-10 triangle.
In a 6-8-10 triangle, the area = (1/2)(6)(8) = 24, which doesn't satisfy the constraint in statement 1 that the area = 25.

Implication:
Only ONE right triangle has a hypotenuse of 10 and an area of 25 -- the same triangle implied by statement 1:
10/√2, 10/√2, 10.
Thus, p = 10/√2 + 10/√2 + 10.
SUFFICIENT.

The correct answer is D.
Last edited by GMATGuruNY on Fri Dec 06, 2013 4:08 pm, edited 1 time in total.
Private tutor exclusively for the GMAT and GRE, with over 20 years of experience.
Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.

As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.

For more information, please email me (Mitch Hunt) at [email protected].
Student Review #1
Student Review #2
Student Review #3
Top

GMAT/MBA Expert

User avatar
GMAT Instructor
Posts: 768
Joined: Wed Dec 28, 2011 4:18 pm
Location: Berkeley, CA
Thanked: 387 times
Followed by:140 members

by Mike@Magoosh » Fri Dec 06, 2013 2:49 pm
Zach.J.Dragone wrote:Thanks for the response Mike!

I am quite familiar with those formulas but I didn't make the connection. I understand we need them (and how the numbers are plugged in) but I guess my problem is why do we need them? How do I know that on the test? I'm not sure that's a question anyone can answer for me...even after doing truck loads of problems I still miss so many because I fail to see these tricks and clues as to what to do next. As soon as I see the explanation (oftentimes simply as soon as I see the correct answer) I know exactly what I did wrong and how to solve the problem, but for some reason I'm just not picking up on all these "signs" in the questions!
Mike@Magoosh wrote: First of all, I am going to recommend a post I think you will find very helpful:
https://magoosh.com/gmat/2013/how-to-do- ... th-faster/
What are asking has very much to do with pattern-matching and the way it plays out in math solutions.
Zach,
Once again, I highly recommend that link. What you are asking sounds to me very much like a left-brain superstar asking what left-brain approach he can use to open up right-brain insights. As you will see in that blog, it's not so straightforward, but it can be done.
Mike :-)
Magoosh GMAT Instructor
https://gmat.magoosh.com/
Top

GMAT/MBA Expert

User avatar
Elite Legendary Member
Posts: 10392
Joined: Sun Jun 23, 2013 6:38 pm
Location: Palo Alto, CA
Thanked: 2867 times
Followed by:511 members
GMAT Score:800

by [email protected] » Fri Dec 06, 2013 4:11 pm
Hi Zach,

There's an error in your math.

If you're going to attempt to answer this question using "substitution", then you're going to need to use the formulas that we KNOW are correct:

X^2 + Y^2 = 100
XY = 50

We DON'T know if X + Y = 10, so you can't rely on that.

Try solving with the above 2 equations and you CAN get the answer; it just takes awhile.

GMAT assassins aren't born, they're made,
Rich
Contact Rich at [email protected]
Image
Top
Post new topic Post Reply
• Page 1 of 1