Is the area of the triangular region above less than 20?

As in the attached picture. Even after combining, area could still be > 20 or < 20. Insufficient.

E answer

Target question: Is the area of the triangular region above less than 20?

I'm going to head straight to....

Statements 1 and 2 combined
Consider the following two scenarios, each of which satisfies BOTH statements.

case a: The triangles is an equilateral triangles and each side has length 0.001. As you can imagine, we need not find the area of this triangle since it's clear that the area is LESS THAN 20

case b: Even though statement 1 tells us that x and y cannot be the legs of a right triangle, let's see what would happen if those sides WERE the legs of a right triangle. Also, let's say that x and y are both equal to 6.5. In this case, the base has length 6.5 and the height has length 6.5. So, the area = (base)/(height)/2 = (6.5)(6.5)/2 = 21.125
Granted this scenario breaks both of the given conditions (x and y ARE the legs of a right triangle AND x+y is NOT less than 13, HOWEVER, we need only recognize that if were were to reduce the angle between x and y from 90 degrees to 89.9999999 degrees, then the area of the triangle would be a teeeeny bit less than 21.125. This would mean that x and y are NOT the legs of a right triangle, so statement 1 is satisfied.
Likewise, if we were to reduce the length of x from 6.5 to 6.49999999, then the area of the triangle would be a teeeeny bit less than 21.125. This would mean that x+y < 13, so statement 2 is satisfied.
As we can see, we can make it so that case b satisfies both conditions, and have it so that the area of the triangle is GREATER THAN 20
Since we cannot answer the target question with certainty, the combined statements are NOT SUFFICIENT

Answer:

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Is the area of the triangular region above less than 20?

1) \(x^2 + y^2≠ z^2\)
\(x^2 + y^2\) , not being equal to \(z^2\)
could mean any of the following :
\(x^2 + y^2 < z^2\)
\(x^2 + y^2 > z^2\)
We cannot clearly ascertain what the values of x and y are.
Therefore, we cannot tell anything about the area of the triangular region. (Insufficient)

2) x + y < 13
If x+y < 13 the sum could be as less as 3 and as big as 12.
x = 1,y=2 will give us an area lesser than 20
x = 5,y=7 will give us an area greater than 20. (Insufficient)

Combining the statements, we still can't clearly tell anything about the area of the triangle
as already explained in Statement 2 (Insufficient - Option E)

Area of triangle= \(\frac{1}{2}*Base * height\)
(1) x^2 + y^2≠ z^2
This option states that the triangle is not right angled
No data on base and height
NS
(2) x + y < 13
Two variable one equation and an inequality
Clearly NS

St 1+2 states no info on base and height

Option E

I thought I'd point out that your rationale to conclude that the combined statement are not sufficient (i.e., "1+2 states no info on base and height") is not quite correct.
If the target question had asked "Is the area less than 22?" (instead of asking "Is the area less than 20?", then statement 2 would be sufficient.

Cheers,
Brent

@Brent, thankyou for your observation on my approach. But I need clarification on the proposed approach.

So if the question had been
\(Area<22\)

B.\(x+y<13\)

Approach 1:
There are two cases


Case 1: Right angle triangle
Case 2: Other triangles (Scalene, equilateral)

Case 1: Assume any values for x,y such that the sum is less than 13,
And the area can be found and it is less than 22.

Case 2: How can I find the area of other triangles.

Should I assume a Value for x and y and get the range of values for z by formula
\(x-y<z<x+y\)
and then I can find height and consider z as base

Approach 2:
Orelse it's safer to find the case for equilateral triangle area, as it has the greatest area of all the triangles

If the question were "Is the area < 22?" , we don't need to do as much work.

(2) x+y<13
Let's just consider a triangle where two of the sides have lengths 6.5 and 6.5
What's the greatest area possible?
Let's make the base have length 6.5
Since area = (base)(height)/2, we will MAXIMIZE the area if we can MAXIMIZE the height.
If the other side has length 6.5, we can maximize the height by making that side PERPENDICULAR to the base, in which case the area = (6.5)(6.5)/2 = 21.125
If x+y = 13, then the MAXIMUM area = 21.25
If x+y < 13, then the MAXIMUM area is less than 21.25
If the maximum area is less than 21.25, then the answer to the target question is "No! The area is definitely less than 22"
So, statement 2 is sufficient.

Cheers,
Brent

1) This statement just tells us that it is not a right angled triangle i.e. x is not the height.

No way we could find the area.

BCE

2) x+y<13

This is fine but how do we know what is the height of this triangle?

Not sufficient

Both

Not a right angled triangle and we can't find the height.


E

Hello Brent

Just by looking at the figure how can we determine which one is the height?



Thanks

Hey Brent,

I think the maximum area, if 2 perpendicular sides totalled 13, would be 21.125

This is not the best explanation:
"x = 5,y=7 will give us an area greater than 20. Insufficient"

For X=5;Y=7, Area will be less than 20 (to be precise - less than or equal to 17.5).

Only for X >= 5.72 and Y = 7, area will be equal to or more than 20

Area of triangular region is the greatest for right triangle:
Formula is: 1/2* X * Y = 1/2*5*7 = 17.5 (less than 20)
For all triangles other than right triangle, area will be even smaller for given X and Y.

Tested using following calculator for all possible angles (1-189):
https://www.google.com.ua/search?q=tria ... F-8&skip=s

Please correct me if I'm wrong.

Hi Brent,

I couldn't understand this part of your explanation "If the other side has length 6.5, we can maximize the height by making that side PERPENDICULAR to the base, in which case the area = (6.5)(6.5)/2 = 21.125". Is area of a triangle highest when the triangle is a right angle triangle? Also, i presume that when two sides are equal and the given triangle is a right triangle, then both 6.5 ought to be legs as third side will be hypotenuse and hence greater than two other sides. Hence, you took 6.5 as a height.

If the lengths of two sides of a triangle are x and y, then we will maximize the area of that triangle if we make those sides PERPENDICULAR to each other.
Why is this?
Since any side of a triangle can be considered the base, let's let the side with length x be the base.
Area of triangle = (base)(height)/2 = (x)(height)/2
To maximize the area, we must now maximize the height. If we make the sides with length x and y PERPENDICULAR to each other, then the height of the triangle will EQUAL y
If the sides with length x and y are NOT perpendicular, then the height of the triangle will be LESS THAN y

Does that help?

Cheers,
Brent

when x and y are two sides of a square angle, the are is biggest.

if x side turn to right or to left, the area is smaller because the side y is the same and h of the triangle is smaller.

if x+y<13 this mean x, y < 6 and 7
6x7=42= tw time the area
so, combine two statement, xy<21
we can not know whether xy<20

E

We need to determine whether the area of the triangular region is less than 20.

Statement One Alone:

x^2 + y^2 ≠ z^2

Because we don’t know anything about the values of x, y, and z, statement one alone is not enough information to answer the question.

Statement Two Alone:

x + y < 13

Again, we can’t determine the values of x and y (and z), so statement two alone is not enough information to answer the question.

Statements One and Two Together:

Using both statements, we still can’t determine the value of x and y (and z). For example, if we take y as the base of the triangle, we see that the height of the triangle, h, has to be less than x. So, let’s say y = 10, x = 2.5, and h = 2; then, the area of the triangle is ½ x 10 x 2 = 10, which is less than 20. However, let’s say y = 6.4, x = 6.5, and h = 6.4; then, the area of the triangle is ½ x 6.4 x 6.4 = 20.48, which is greater than 20.

Answer: E
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Very nice approach to find the max area of the triangle !!!

Dear Experts,
So can i generally infer that, the max area of a triangle with sides x,y (such that x+y<13) will be a right angled isosceles triangle such that x=y.
Thus x+x<13 and Area<1/2*(6.5)x(6.5)
i.e Area <21.125

Think about why it is so in the above example of x+y<13
For a given sum of two numbers, their product is maximized when they are equal.
That is the reason he took the example of x=6.5 and y=6.5 to find what the largest possible area could be.

Hi all. Good to see some good discussion on this question. ?
Let me put it together in my way.
One general point of the question is to not trust the figure on DS questions.

You, definitely, do not need to plug in for statement (1). It just states indirectly that the triangle should not be right angled.
So, insufficient.
Left with BCE.

The second statement states that the sum of x and y is less than 13. If the number would have been bigger (like 100), the problem would have been much easier and would not have required any plugging in also. But, since that is not the case and it’s a GMAT problem, we need to do PI.

x = 6 and y = 7 and consider the triangle to be right angled. What do you get? Area = 21. Since the sum of x and y should be less than 13, area will be less than 21. But, it could be 5 as well as 20.5.
So, insufficient.
Left with CE.

Take the same case as used in the second statement. Since the triangle is not right angled, we know that the answer should be less than 21. Again, it could be more than 20 or less than 20.
Insufficient.

So, the answer is E.

Point — In some geometry questions involving inequalities, it makes sense to solve the case which replace inequality sign by equation and then compare.