Question

question 10 (multiple choice worth 1 points) (04.01 lc) which statement explains how you could use coordinate geometry to prove that quadrilateral abcd is a square? prove that segments ad and ab are congruent and parallel. prove that all sides are congruent and that the slopes of consecutive sides are opposite reciprocals. 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:

To prove a quadrilateral is a square, we need to show all sides are congruent (equal in length) and consecutive sides are perpendicular (slopes are negative reciprocals, product of slopes = -1). Let's analyze each option:

• Option 1: Proving AD and AB congruent and parallel would show a parallelogram, not necessarily a square (could be a rhombus or rectangle).
• Option 2: Proving all sides congruent (so it's a rhombus) and slopes of consecutive sides are opposite reciprocals (so consecutive sides are perpendicular, hence angles are right angles) proves it's a square (a rhombus with right angles).
• Option 3: Proving opposite sides congruent and slopes of consecutive sides equal would mean consecutive sides are parallel, which is a parallelogram, not a square.
• Option 4: Proving BC and CD congruent and parallel shows a parallelogram, not sufficient for a square.

So the correct option is the one that proves all sides congruent and consecutive sides' slopes are opposite reciprocals.

Answer:

The option "Prove that all sides are congruent and that the slopes of consecutive sides are opposite reciprocals." (the third option in the list of choices)