Question

If three consecutive vertices of a parallelogram are \( A(1,-2) \), \( B(3,6) \) and \( C(5,10) \), find its fourth vertex.

Hint:

If three consecutive vertices \( A \), \( B \), and \( C \) of a parallelogram are known, then the fourth vertex is found by \( D = A + C - B \).

Answer 1

Step 1: Use the property of a parallelogram.}
In a parallelogram, if \( A \), \( B \), and \( C \) are three consecutive vertices, then the fourth vertex \( D \) is obtained by using the vector relation \[ \vec{D} = \vec{A} + \vec{C} - \vec{B} \]
Step 2: Write the coordinates of the given vertices.}
The given points are \[ A(1,-2), \quad B(3,6), \quad C(5,10) \]
Step 3: Apply the coordinate formula for the fourth vertex.}
Using \[ D(x,y) = (x_A + x_C - x_B,\; y_A + y_C - y_B) \] we get \[ D = (1 + 5 - 3,\; -2 + 10 - 6) \]
Step 4: Simplify the coordinates.}
Now simplify each coordinate: \[ x = 1 + 5 - 3 = 3 \] \[ y = -2 + 10 - 6 = 2 \] So, \[ D = (3,2) \]
Step 5: Write the final answer.}
Hence, the fourth vertex of the parallelogram is \[ \boxed{(3,2)} \]