Question: The coordinates of vertices P and Q of an equilateral triangle PQR are (–4, 0) and (4, 0) respectively. Which of the following could be the coordinates of vertex R?

1. (0,2√3)
2. (0,4)
3. (0,4√3)
4. (4√3,0)
5. (4,4√3)

Solutions and Explanation

Approach Solution : 1

Calculations are not necessary to arrive at a logical solution to this problem.,
It is given that the triangle is equilateral, P=(-4, 0) and Q = (4, 0)
Now to determine coordinates of R
It follows that P and Q are on the X-axis and are separated from the Origin O by a distance of 4 units.
We are aware that PQR is an equilateral triangle, so the other vertex R must be located above the midpoint of PQ, which is on the Y axis.
This removes the fourth and fifth options respectively
The points ORQ can be used to create a right triangle measuring 30-60-90.
The smallest side OQ is on the other side of the smallest angle that is 30°.
The smallest side OQ is on the other side of the smallest angle that is 30°. As a result, the distance between the origin and Point R must be greater than 4.

This removes the first and second options respectively.

Correct Answer: (C)

Approach Solution : 2

The formula that can be used to determine how far any two points (vertices in this case) are apart is{(x2-x1)2+(y2-y2)2}

\(\sqrt{(x2-x1)^2+(y2-y2)^2}\)
 

With the provided vertices, we can determine that their separation is 8 units (plug numbers into the formula).
Due to the fact that the triangle is equilateral, the third vertex will be 8 units away from any of the given vertices.
Here, we can just enter some numbers and find which will give output as 8 units.
From the given options, only (0,4√3) gives the required outpu.

Correct Answer: (C)

“The coordinates of vertices P and Q of an equilateral triangle PQR are” - is a subject that the GMAT quantitative reasoning section covers. A school needs to be proficient in a variety of qualitative skills to answer the GMAT Problem Solving questions correctly. There are 31 questions in the GMAT Quant section as a whole. The GMAT Quant topics' problem-solving section calls for the resolution of calculative mathematical puzzles.

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