Brent@GMATPrepNow wrote:[email protected] wrote:If the vertices of a triangle have coordinates (x,1), (5,1), and (5,y) where x<5 and y>1, what is the area of the triangle?
(1) x=y
(2) Angle at the vertex (x,1) is equal to angle at the vertex (5,y)
Target question: What is the area of the triangle?
Given: The vertices of a triangle have coordinates (x,1), (5,1), and (5,y)
Let's first examine the point (x,1)
This tells us that the y-coordinate is fixed at 1, but the x coordinate is not fixed.
So, the point (x, 1) can be anywhere on the green line (below)
Likewise, the point (5, y) has its x-coordinate fixed at 5, but the y-coordinate can have many values.
So, the point (5, y) can be anywhere on the red line (below)
Now let's add the point (5,1)
Finally, we're told that x < 5 and y > 1.
So, the x-coordinate on the green line must be less than 5, and the y-coordinate on the red line must be less than 5.
Once we restrict the x- and y-coordinates, we can see that the points (x, 1) and (5, y) lie on the green and red lines respectively.
Okay, now it's time to examine the statements . . .
Statement 1: x = y
It will help to take a look at all points such that the x- and y-coordinates are equal.
To do so, let's graph the line y = x (in purple below)
Every point on the purple line satisfies the condition that x = y.
Since our the two remaining points of our triangle must also be on the green and red lines, we can see that these points must lie at the intersections of the purple lines and the green and red lines.
As you can see, this creates ONE AND ONLY ONE triangle, which means
there's only one possible area for the triangle. (Aside: We don't need to actually find this area. We need only determine that there can be only one area).
Since we can answer the
target question with certainty, statement 1 is SUFFICIENT
Statement 2: The angle at the vertex (x,1) is equal to angle at the vertex (5,y)
All this tells us is that the two remaining angles in the right triangle must both equal 45 degrees.
So, there are many different ways that this can happen.
As we can see, there are
many possible triangles with different areas.
Since we cannot answer the
target question with certainty, statement 2 is NOT SUFFICIENT
Answer =
A
Cheers,
Brent
