Question

james is writing a coordinate proof involving a parallelogram. knowing that the opposite sides of a parallelogram are congruent, james places his parallelogram on the coordinate plane such that one vertex is at the origin and one side lies along the x - axis. what coordinates should he assign to the fourth vertex of the parallelogram? (\\(\frac{a}{2},0\\)) (\\(2a,\frac{a}{2}\\)) (\\(2a,a\\)) (\\(2a,2a\\))

Explanation:

Step1: Use the property of parallelogram

In a parallelogram, opposite sides are parallel and equal. If one side is from \((0,0)\) to \((a,0)\) (length \(a\) along \(x\)-axis), and another side is from \((0,0)\) to \((a,a)\).

Step2: Find the fourth vertex

The vector from \((0,0)\) to \((a,a)\) is \(\langle a,a
angle\), and the vector from \((0,0)\) to \((a,0)\) is \(\langle a,0
angle\). To get the fourth vertex, we add the vector \(\langle a,a
angle\) to the point \((a,0)\).
The \(x\)-coordinate: \(a + a=2a\)
The \(y\)-coordinate: \(0 + a=a\)

Answer:

\((2a,a)\) (the third option)