gmatt1476 wrote:
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
HSince the red triangle is an isosceles triangle, we know that base of the triangle must also be
HThis means the length of side XZ =
B +
HAt 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: 2
B + 2
H = 24
Divide both sides by 2 to get:
B +
H =
12This means the length of side XZ =
B +
H =
12Also, since triangle XYZ is an isosceles triangle, we know that side YZ is also
12So 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√2Since we can answer the
target question with certainty, the combined statements are SUFFICIENT
Answer: C
Cheers,
Brent