A certain right triangle has sides of length x, y, and z, where x

Question

A certain right triangle has sides of length x, y, and z, where x < y < z. If the area of this triangular region is 1, which of the following indicates all of the possible values of y?

A.  y > \sqrt {2}

B.  \frac {\sqrt {3}} {2} < y < \sqrt {2}

C.  \frac {\sqrt {2}} {3} < y < \frac {\sqrt {3}} {2}

D.  \frac {\sqrt {3}} {4} < y < \frac {\sqrt {2}} {3}

E.  y < \frac {\sqrt {3}}{4}

Bunuel · Accepted Answer

The area of the triangle is \frac{xy}{2}=1 ( x x=\frac{2}{y} . As x 2 \sqrt{2} x=\frac{2}{1,000,000} . Hope it helps.

KarishmaB · Answer

You can think about it in terms of transition points too.

x < y< z so this means that x and y are the legs of the right triangle and z is the hypotenuse.
The area of the triangle will be (1/2)*xy = 1
xy = 2

Now, if x = y, then both x and y would be equal to  \sqrt{2} .
But y is greater than x, so y would be at least a slight bit greater than  \sqrt{2}  and x would be a slight bit less than  \sqrt{2} . In all options other than (A), y takes values less than 1.414.
Answer (A)

BrentGMATPrepNow · Answer

Area of triangle  = \frac{base 	imes height}{2}

There are   infinitely many   right triangles that have an area of 1. 
So, one approach is to find a triangle that meets the given conditions, and then see what conclusions we can draw.

Here's one such right triangle: 
 https://i.imgur.com/0w5pyi1.png 
 Aside: There's no need to calculate the actual value of z (which is √5), since the question isn't asking us about the possible values of z.

This triangle meets the condition that its area is 1, and that x < y < z 
With this particular triangle,   y = 2

When we check the answer choices, only one option (answer choice A) allows for y to equal 2

Answer: A

Cheers,
Brent

rohansharma · Answer

Dear Bunuel,

While solving the question ,I assumed it to be the special 90,60,30 triangle.
Am I wrong in following that approach ?

Bunuel · Answer

We are just told that the triangle is right, not that it's a special kind like 30-60-90 or 45-45-90.

subhajeet · Answer

OA is A
since this is a rt angled triangle so z is th hypotnuse
and given xy = 2
so as x decreased y increases. Now if x is 1 then y is 2, when x is 1/2 y is 4.
Only option A supports this result.

2flY · Answer

Could you explain why that is?

Bunuel · Answer

Just substitute x with  \frac{2}{y}  in  x<y  to get  \frac{2}{y}<y .

WoundedTiger · Answer

Hi Bunuel,

The Q stem says that sides are x<y<z and it is a right angle triangle. So we can assume that it will be 30-60-90 triangle. Had it been 45-45-90 triangle then the 2 sides ie base and perpendicular would have been equal and therefore x=y and x,y<z

I guess it should be okay to assume that it is 30-60 -90 triangle.

Please confirm

Bunuel · Answer

Yes, if it were 45-45-90, then we would have that x=y<z. BUT, knowing that it's not a 45-45-90 right triangle does NOT mean that it's necessarily 30-60-90 triangle: there are numerous other right triangles. For example, 10-80-90, 11-79-90, 25-65-90, ...

Hope it's clear.

Dienekes · Answer

One other way that I noticed to solve this problem is to check the length of  y  when  x=y , i.e. 45,45,90. In that case  x=y=\sqrt{2} , however as  y>x , it'd always need to be  >\sqrt{2} .

HTH

ENGRTOMBA2018 · Answer

First step will be to breakdown the options into recognizable decimal representations (assuming \sqrt{2} \approx 1.4 , \sqrt{3} \approx 1.7 ) A) y>1.4 B) 0.8 x*y=2 Now from the relation xy=2 --> go back to the options and test for y=1. You get x=2 but we are given that x y MUST be > \approx 1.4 such that x < y For any value of y < 1.4 , you will end up getting x>y (try with y=0.5 or 0.75 etc). Only A satisfies this condition and is hence the correct answer. Hope this helps.

KarishmaB · Answer

Since it is a right triangle, z is the greatest side and hence the hypotenuse.

So area of the triangle will be (1/2)*xy = 1
xy = 2

Note that x < y.
If x were equal to y,  x = y = \sqrt{2} 
Since x is less than y,  y > \sqrt{2}  and  x < \sqrt{2}

Answer (A)

BrentGMATPrepNow · Answer

If x < y < z, then our RIGHT triangle looks something like this (where the hypotenuse is always the longest side)
 https://i.imgur.com/PX59f5j.png

When we scan the answer choices ( ALWAYS scan the answer choices before performing any calculations! ), I see that B, C, D and E all provide a MAXIMUM value of y. 
This should be a bit of a surprise, because there's no limit to the length of each leg of the triangle.

GIVEN: The triangle has area 1 
Area = (base)(height)/2
So, here's one possible triangle with area 1: 
 https://i.imgur.com/yNaonqh.png 
ASIDE: Area = (10)(0.2)/2 = 2/2 = 1 (voila!)

In this triangle, x = 0.2, y = 10 and z = some number greater than 10
Since it's possible for y to equal 10, we can eliminate answer choices B, C, D and E

Answer: A

RELATED VIDEO 
 https://www.youtube.com/watch?v=iUUP2sp4ogg

mimajit · Answer

Since it is a right triangle, z is the greatest side and hence the hypotenuse.

So area of the triangle will be (1/2)*xy = 1
xy = 2

Note that x < y.
so for XY to be 2 either x=2 and Y=1 or Y =2 and X =1

we are already told x < y. so Y has to be 2.... and Root2 = 1.4 and since we now know y =2 option A

Kinshook · Answer

Given: A certain right triangle has sides of length x, y, and z, where x < y < z. Asked: If the area of this triangular region is 1, which of the following indicates all of the possible values of y? Area of right triangle = xy/2 = 1 xy = 2 x = 2/y < y y^2 >2 y > \sqrt{2} IMO A

rahulbhusan · Answer

Given, the area of this triangular region is 1 and x < y < z.
So, area of triangle= 1/2*x*y ( as it's a right angled triangle, so z must be the hypotenuse, and x and y can be either base or height)
=> 1/2*x*y=1
=> x*y=2
now, lets find the minimum value for y (or maximum value of x).
As, its a right angled triangle, the maximum value of x will be when its a isosceles right triangle i.e. 45:45:90 triangle.
=> x=y<z
so, x*y=2
=>x*x=2 (i.e minimum value of y)
=> x^2=2
=> x=√2
But, as we know that y>x,
so, y>√2. Option A

Basshead · Answer

A certain right triangle has sides of length x, y, and z, where x < y < z. If the area of this triangular region is 1, which of the following indicates all of the possible values of y?

We can approximate:
\(\sqrt{2} = 1.4\)
\(\sqrt{3} = 1.7\)

If the area of the triangular region = 1, this means \(\frac{xy}{2} = 1\)

A) \(y > 1.4\)

B) \(0.7 < y < 1.4\)

C) \(\frac{1.4}{3} < y < \frac{1.7}{2}\)

D) \(\frac{1.7}{4} < y < \frac{1.4}{3}\)

E) \(y < \frac{1.7}{4}\)

The only range that works is \(y > 1.4\). Answer is A.

anniaustin · Answer

xy=2 ,
or,  x=\frac{2}{y} 
 x<y 
or,  \frac{2}{y}<y 
or,  2<y^2 
or,  y>2

Answer: A

A certain right triangle has sides of length x, y, and z, where x < y

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

A certain right triangle has sides of length x, y, and z, where x < y

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards