Question: 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. Angles at the vertex (x,1) is equal to the angle at the vertex (5,y)

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are NOT sufficient.

“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?” – is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken f0rom the book "GMAT Quantitative Review". GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

Answer:

Approach Solution 1:

Firstly look at the given diagram, we can notice that vertex (x,1) will be somewhere on the green line segment and the vertex (5,y) will be on the blue line segment. So, in any case our triangle will be right angled, with a right angle at the vertex (5,1). Next, the length of the leg on the green line segment will be 5-x and the length of the leg on the blue line segment will be y-1.

So the area of the triangle will be: \(\frac{1}{2}*(5-x)*(y-1)\)

Considering Statement (1), we have: x = y
Since x < 5 and y > 1 then both x and y are in the range (1,5): 1 < (x = y) < 5.

If we substitute y with x, we will get: area = \(\frac{1}{2}*(5-x)*(y-1)\)  = \(\frac{1}{2}*(5-x)*(x-1)\) , different values of x give different values for the area (even knowing that 1 < x < 5).
Hence, this statement is insufficient.

Now consider Statement (2), we have: Angle at the vertex (x,1) is equal to the angle at the vertex (5,y)
We have isosceles right triangle: 5-x = y-1

Again, if we substitute y-1 with 5-x, we will get: area = \(\frac{1}{2}*(5-x)*(y-1)\) = \(\frac{1}{2}*(5-x)*(5-x)\) , different values of x give different values for the area.
Hence this statement is also insufficient.

Now combining both the statements (1) + (2), we will get that x = y and 5 – x = y – 1

Solve for x: x = y =3, so area = \(\frac{1}{2}*(5-3)*(3-1)\) = 2
Hence this is sufficient.

Correct option: C

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