Question

Which statement explains how you could use coordinate geometry to prove that quadrilateral abcd is a square? shape abcd is shown. point a is at negative 3, 5. point b is at 1, 7. point c is at 3, 3. point d is at negative 1, 1. 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

Answer 1

To prove a quadrilateral is a square with coordinate geometry, calculate all side lengths to ensure they are congruent and use slopes to show that consecutive sides are perpendicular, with opposite reciprocals as slopes.

Step-by-step explanation:

To prove that the quadrilateral ABCD is a square using coordinate geometry, you can use the distance formula to show that all sides are congruent (have equal length) and use the slope formula to demonstrate that consecutive sides are perpendicular to each other (the slopes of consecutive sides are opposite reciprocals). For a square, not only do all sides need to be congruent, but consecutive sides must also form right angles, hence the perpendicularity requirement. If AB is perpendicular to BC, BC is perpendicular to CD, CD is perpendicular to DA, and DA is perpendicular to AB, and all the sides are equal in length, then the quadrilateral is a square.

Here are the steps you would follow:

• Use the distance formula to calculate the length of each side: AB, BC, CD, and DA. The distance formula for two points (x1, y1) and (x2, y2) is √((x2 - x1)² + (y2 - y1)².
• Use the slope formula to determine the slope of each side. The slope formula for two points (x1, y1) and (x2, y2) is (y2 - y1) / (x2 - x1).
• Verify that each pair of consecutive sides are perpendicular, meaning the product of their slopes is -1. This shows the sides form right angles with each other.
• Once you've established that all sides are congruent and form right angles with each other, you have proven the quadrilateral is a square.

It is not sufficient to only prove that segments AD and AB are congruent and parallel, or that opposite sides are congruent and that the slopes of consecutive sides are equal, or that segments BC and CD are congruent and parallel, as these conditions alone do not meet all the criteria for a square.

Answer 2

As a result of AC and BD's zero dot product, the diagonals are perpendicular. Therefore , ABCD is a square.

How to determine this

You can use coordinate geometry to demonstrate that the quadrilateral ABCD is a square by doing the following steps:

Determine the separations between the points: To find the lengths of the quadrilateral's four sides, use the distance formula. ABCD is a rhombus if the lengths of all four sides are equal.

Determine the side slopes: The quadrilateral's four side slopes can be determined using the slope formula. ABCD is a rectangle if the slopes of the successive sides are negative reciprocals of each other and the slopes of the opposite sides are equal.

Verify if the diagonals are perpendicular. ABCD is a square if both of the quadrilateral's diagonals are perpendicular.

The coordinates of the four points in this instance are as follows: A (-3, 5), B (1, 7), C (3, 3), and D (-1, 1).

Calculating distances:

AB = √((1 - (-3))^2 + (7 - 5)^2) = √(16 + 4) = √20

BC = √((3 - 1)^2 + (3 - 7)^2) = √(4 + 16) = √20

CD = √((-1 - 3)^2 + (1 - 3)^2) = √(16 + 4) = √20

DA = √((-3 - (-1))^2 + (5 - 1)^2) = √(4 + 16) = √20

Since all four sides have the same length, ABCD is a rhombus.

Calculating slopes:

m_AB = (7 - 5) / (1 - (-3)) = 2/4 = 1/2

m_BC = (3 - 7) / (3 - 1) = -4/2 = -2

m_CD = (1 - 3) / (-1 - 3) = -2/4 = -1/2

m_DA = (5 - 1) / (-3 - (-1)) = 4/2 = 2

Although the slopes of successive sides are not negative reciprocals of one another, the slopes of opposite sides are equal (m_AB = m_CD and m_BC = m_DA). ABCD is not a rectangle as a result.

Verifying if the diagonals are perpendicular:

You can compute the dot product of the vectors representing the diagonals to see if they are perpendicular. In the event that the diagonals are perpendicular, the dot product will be 0.

Vector AC = (3 - (-3), 3 - 5) = (6, -2)

Vector BD = (1 - (-1), 7 - 1) = (2, 6)

Dot product of AC and BD = (6 * 2) + (-2 * 6) = 0