Question

In the figure above, ABCE is a square. What are the coordinates of point B? 

 

• A. (-4,2)
• B. (-2,4)
• C. (-2,6)
• D. (4,-6)
• E. (6,-2)

Hint:

Another method is to use slopes. Find the slope of AC (\(m_{AC}\)). The slope of the other diagonal, BE, will be the negative reciprocal (\(m_{BE} = -1/m_{AC}\)). You know the midpoint of BE is (-1,1). You can write the equation for the line BE and check which of the options for B lies on that line.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This is a coordinate geometry problem involving the properties of a square. We are given the coordinates of two opposite vertices (A and C) and need to find the coordinates of one of the other vertices (B).
Step 2: Key Formula or Approach:
A key property of a square is that its diagonals are perpendicular and bisect each other. We can use this property:
1. Find the midpoint of the given diagonal AC. This point is also the midpoint of the other diagonal BD.
2. Find the vector from the midpoint to the known vertex C.
3. Rotate this vector by 90° counter-clockwise to find the vector from the midpoint to the vertex B.
4. Add this vector to the midpoint's coordinates to find the coordinates of B.
Step 3: Detailed Explanation:
We are given the coordinates A(-6, 0) and C(4, 2).
1. Find the midpoint of AC:
The midpoint \(M\) of a segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\) is \((\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})\).
\[ M = \left(\frac{-6+4}{2}, \frac{0+2}{2}\right) = (-1, 1) \]
2. Find the vector from M to C:
A vector from point P to point Q is given by (Q_x - P_x, Q_y - P_y).
Vector \(\vec{MC} = (4 - (-1), 2 - 1) = (5, 1)\).
3. Rotate the vector \(\vec{MC}\) by 90° counter-clockwise:
A rotation of a vector \((x,y)\) by 90° counter-clockwise gives the vector \((-y, x)\).
Rotating \(\vec{MC} = (5, 1)\) gives:
\[ \vec{MB} = (-1, 5) \]
4. Find the coordinates of B:
The coordinates of B are found by adding the vector \(\vec{MB}\) to the midpoint M:
\[ B = M + \vec{MB} = (-1, 1) + (-1, 5) = (-2, 6) \]
Step 4: Final Answer:
Using the properties of the diagonals of a square, the coordinates of B are (-2, 6).