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)
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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
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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
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Hello Brent

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



Thanks

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.

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
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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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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
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