In the figure above, if the shaded region is rectangular, what is the

Target question: What is the length of XY ?

Statement 1: The perimeter of the shaded region is 24
For the statement we need only recognize that there are infinitely many right triangles in which the same rectangle (with area 24) can be inscribed.
Here are two examples:

As you can see, the length of XY varies, which means we can't answer the target question with certainty.
Statement 1 is NOT SUFFICIENT

Statement 2: The measure of \(\angle XYZ\) is 45°
Since we don't know any lengths in the diagram, there are infinitely many sizes of diagrams that meet statement 2.

In all of the above examples, angle XYZ is always 45°, yet the length of XY varies, which means we can't answer the target question with certainty.
Statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
If ∠XYZ = 45°, we can also conclude that ∠YXZ = 45° (since the three angles in any triangle always add to 180 degrees)
So, let's add all of the 45° angles in the diagram.
At the same time let's let B = the length of the rectangle's base, and let H = the length of the rectangle's height


From here, let's focus on the red triangle below.

The red triangle has height H
Since the red triangle is an isosceles triangle, we know that base of the triangle must also be H
This means the length of side XZ = B + H

At this point let's use the information from Statement 1, which tells us that the perimeter of the shaded rectangle is 24
This means we can write B + H + B + H = 24
When we simplify this we get: 2B + 2H = 24
Divide both sides by 2 to get: B + H = 12

This means the length of side XZ = B + H = 12
Also, since triangle XYZ is an isosceles triangle, we know that side YZ is also 12
So we have the following:


At this point we can find the length of XY by using Pythagorean theorem or by comparing it to the base 45-45-90 right triangle.
Either way, we'll find that side XY has length 12√2
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer: C

Cheers,
Brent

In the figure above, if the shaded region is rectangular, what is the length of XY?

STATEMENT (1) The perimeter of the shaded region is 24.
--let the length of the rectangle be = l and breadth be = b
from this statement we get l+b = 12
from this, we cant find length of YZ and XZ
hence we can't find the value of XY
INSUFFICIENT

STATEMENT (2) The measure of ∠XYZ is 45°
from this, we know the triangle is isosceles but still, we don't know anything about sides.
we cant find the value of XY
INSUFFICIENT

combining both statements
we know the triangle is isosceles and l+b = 12
from this, we can find the value of YZ and XZ
and hence we can find value of XY
SUFFICIENT

C is the correct answer

IMO and is c, given p=24 so 2(l+b)=(24,l+b=12,given y=45 degrees, so its a 45-45-90 triangle , two small triangles are also 45-45-90 triangles so length of xy=√2(l+b)=12√2

Posted from my mobile device

Quick question!

can we take it for granted that such a triangle is a right angle when only one angle of 45 degrees is given?

You're correct to say that, if we only know one angle in a triangle, then we can't conclude that the triangle is a right triangle.
However, in this case, since we're told the shaded region is a rectangle, so, before we even examine any statements, we can already be certain that angle XZY is 90 degrees.
So, once we add statement 2 (telling us angle XYZ is 45 degrees), we now know TWO angles.

IMO and is c, given p=24 so 2(l+b)=(24,l+b=12,given y=45 degrees, so its a 45-45-90 triangle , two small triangles are also 45-45-90 triangles so length of xy=√2(l+b)=12√2

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.
_________________