Question

question 5 (multiple choice worth 1 points) (04.01rlc) which statement explains how you could use coordinate geometry to prove that quadrilateral abcd is a square? prove that all sides are congruent, and the slopes of consecutive sides are opposite reciprocals prove that segments ad and ab are congruent and parallel prove that opposite sides are congruent and that the slopes of consecutive sides are equal prove that segments bc and cd are congruent and parallel

Explanation:

A square has all sides congruent (equal in length). Also, adjacent sides are perpendicular. In coordinate geometry, the slope of a line is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). If two lines are perpendicular, the product of their slopes is \(- 1\), which means the slopes of consecutive sides (adjacent sides) are opposite reciprocals (\(m_1\times m_2=-1\)). Proving all sides are congruent (using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)) and slopes of consecutive sides are opposite reciprocals (using the slope formula) is a valid way to prove a quadrilateral is a square.

• Proving only two segments \(AD\) and \(AB\) are congruent and parallel is not sufficient as it could be a rhombus (if only two adjacent sides are considered in a wrong way, but in general, a square needs all - side and right - angle properties).
• If the slopes of consecutive sides are equal, the sides are parallel, not perpendicular. A square needs perpendicular adjacent sides.
• Proving only two segments \(BC\) and \(CD\) are congruent and parallel is not sufficient as it does not account for all sides and the right - angle property.

Answer:

Prove that all sides are congruent, and the slopes of consecutive sides are opposite reciprocals.