For any triangle T in the xy–coordinate plan, the center of T is defined to be the point whose x–coordinate...

1. Translate the problem requirements: We need to find the third vertex of a triangle where we know two vertices at \((0,0)\) and \((6,0)\), and we know the center is at \((3,2)\). The center's coordinates are the averages of all three vertices' coordinates.
2. Set up the averaging equations: Use the definition that center coordinates equal the average of vertex coordinates to create two equations - one for x-coordinates and one for y-coordinates.
3. Solve for the unknown vertex coordinates: Use algebraic manipulation to isolate the unknown vertex coordinates from the averaging equations.
4. Verify the solution: Check that our found vertex produces the given center when averaged with the other two vertices.

Execution of Strategic Approach

1. Translate the problem requirements

Let's understand what we're being asked to find. We have a triangle with three corners (vertices), but we only know where two of them are located. We know:

• First vertex is at point \((0,0)\) - this is the origin
• Second vertex is at point \((6,0)\) - this is 6 units to the right on the x-axis
• The "center" of the triangle is at point \((3,2)\)
• We need to find where the third vertex is located

The key insight is understanding what "center" means here. The problem tells us that the center's x-coordinate is the average of all three x-coordinates of the vertices, and the center's y-coordinate is the average of all three y-coordinates of the vertices.

Think of it like this: if you had three friends standing at the three corners of a triangle, the center would be the balance point - where their average position would be.

Process Skill: TRANSLATE

2. Set up the averaging equations

Now let's work with the averaging concept step by step. Let's call our unknown third vertex \((x, y)\).

For the x-coordinate of the center:

We know the center's x-coordinate is 3. This must equal the average of the three x-coordinates:

Average of x-coordinates = \(\frac{0 + 6 + x}{3} = 3\)

For the y-coordinate of the center:

We know the center's y-coordinate is 2. This must equal the average of the three y-coordinates:

Average of y-coordinates = \(\frac{0 + 0 + y}{3} = 2\)

So we have two equations:

\(\frac{0 + 6 + x}{3} = 3\)

\(\frac{0 + 0 + y}{3} = 2\)

3. Solve for the unknown vertex coordinates

Let's solve each equation separately.

Finding the x-coordinate:

\(\frac{0 + 6 + x}{3} = 3\)

\(\frac{6 + x}{3} = 3\)

Multiply both sides by 3: \(6 + x = 9\)

Subtract 6 from both sides: \(x = 3\)

Finding the y-coordinate:

\(\frac{0 + 0 + y}{3} = 2\)

\(\frac{y}{3} = 2\)

Multiply both sides by 3: \(y = 6\)

Therefore, our third vertex is at the point \((3, 6)\).

4. Verify the solution

Let's double-check our answer by confirming that the three vertices \((0,0)\), \((6,0)\), and \((3,6)\) indeed have a center at \((3,2)\).

Check x-coordinate of center:

Average = \(\frac{0 + 6 + 3}{3} = \frac{9}{3} = 3\) ✓

Check y-coordinate of center:

Average = \(\frac{0 + 0 + 6}{3} = \frac{6}{3} = 2\) ✓

Perfect! Our center is indeed \((3,2)\) as required.

Looking at the answer choices, \((3,6)\) corresponds to choice B.

Final Answer

The coordinates of the remaining vertex are \((3,6)\).

Answer: B. \((3,6)\)

Common Faltering Points

Errors while devising the approach

1. Misunderstanding the definition of "center"
Students might confuse the center described in this problem with other geometric center concepts they've learned before (like circumcenter, incenter, or centroid). The problem clearly defines center as the point where coordinates are arithmetic means of vertex coordinates, but students may try to apply formulas from other center definitions, leading them down the wrong path entirely.

2. Setting up the wrong coordinate system or equations
Some students might get overwhelmed by the coordinate geometry aspect and try to use distance formulas, midpoint formulas, or slope calculations instead of recognizing this is purely an averaging problem. They might think they need to find equal distances or use more complex geometric relationships when the solution only requires basic arithmetic mean calculations.

Errors while executing the approach

1. Arithmetic errors in the averaging equations
When solving \(\frac{6 + x}{3} = 3\), students might make basic algebraic mistakes like forgetting to multiply both sides by 3, or incorrectly calculating \(6 + x = 9\) to get \(x = 6\) instead of \(x = 3\). Similarly, for the y-coordinate equation \(\frac{y}{3} = 2\), they might forget to multiply by 3 and leave \(y = 2\).

2. Mixing up x and y coordinates in the equations
Students might accidentally use the wrong coordinates when setting up their averaging equations. For example, they might include a y-coordinate value when calculating the x-coordinate average, or vice versa, leading to incorrect equations and wrong final coordinates.

Errors while selecting the answer

1. Switching the x and y coordinates in the final answer
After correctly calculating \(x = 3\) and \(y = 6\), students might accidentally write the answer as \((6,3)\) instead of \((3,6)\). This coordinate reversal is a common mistake when students are rushing or not carefully tracking which value corresponds to which coordinate throughout their work.