Question

(04.01r lc)
how could you use coordinate geometry to prove that \\( \overline { b c } \\| \overline { a d } \\) ?

prove the slopes are the same.

prove the slopes are opposite reciprocals

prove the lengths are the same.

prove the midpoints are the same

Explanation:

Step1: Recall the property of parallel lines

In coordinate geometry, if two lines are parallel, then their slopes are equal. The formula for the slope \(m\) of a line passing through two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(m=\frac{y_2 - y_1}{x_2 - x_1}\).

Step2: Analyze each option

• Option 1: Prove the slopes are the same

If we calculate the slope of \(\overline{BC}\) using the coordinates of points \(B\) and \(C\), and the slope of \(\overline{AD}\) using the coordinates of points \(A\) and \(D\), and show that these two slopes are equal, then by the property of parallel lines in coordinate geometry (\(m_1 = m_2\Rightarrow l_1\parallel l_2\)), we can prove \(\overline{BC}\parallel\overline{AD}\).

• Option 2: Prove the slopes are opposite reciprocals

If the slopes of two lines are opposite - reciprocals (\(m_1=-\frac{1}{m_2}\)), then the two lines are perpendicular, not parallel.

• Option 3: Prove the lengths are the same

Proving that the lengths of two line segments (\(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)) are equal only shows that the line segments are congruent, not necessarily parallel. For example, a rhombus has all - sides equal but adjacent sides are not parallel.

• Option 4: Prove the midpoints are the same

Proving that the midpoints (mid - point formula \(M=(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\)) of two line segments are the same only shows that the two line segments bisect each other (if we consider two line segments in a plane), but not that they are parallel. For example, the diagonals of a parallelogram bisect each other, but the diagonals are not parallel.

Answer:

Prove the slopes are the same.