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

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In the figure above, if the shaded region is rectangular, what is the [#permalink]
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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

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Re: In the figure above, if the shaded region is rectangular, what is the [#permalink]
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BrentGMATPrepNow
gmatt1476

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

(1) The perimeter of the shaded region is 24.
(2) The measure of $$\angle XYZ$$ is 45°.
Attachment:
2019-09-22_0545.png

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

Cheers,
Brent

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?
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Re: In the figure above, if the shaded region is rectangular, what is the [#permalink]
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Jaya6

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.
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Re: In the figure above, if the shaded region is rectangular, what is the [#permalink]
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
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