Coordinate Geometry Basics: Rectangles and Parallelograms Using Slopes and Distances

1) Plot four points and connect in order: why vertex order matters

When you are given four points, you do not automatically have a rectangle or parallelogram. You have four labeled locations; the shape depends on the order in which you connect them. In coordinate geometry, we typically connect points in the stated order (for example, A→B→C→D→A). A different order can create a different quadrilateral, including a self-crossing shape.

Example: same points, different shapes

Suppose the points are A(0,0), B(4,0), C(4,3), D(0,3).

• Order A→B→C→D→A draws a rectangle.
• Order A→C→B→D→A draws crossing diagonals first, producing a self-intersecting quadrilateral (not a standard rectangle/parallelogram drawing).

Practical tip: If the problem says “connect in order,” follow that. If it asks whether the four points “can form” a rectangle/parallelogram, you may need to choose an order that makes a simple (non-crossing) quadrilateral.

How to choose a reasonable order (when not given)

• Plot the points and visually identify the “around the outside” order (clockwise or counterclockwise).
• A quick coordinate method: find the centroid estimate ((x1+x2+x3+x4)/4, (y1+y2+y3+y4)/4), then order points by angle around that center (conceptually: go around the outside).

2) Use slopes to test parallel and perpendicular sides

Rectangles and parallelograms are identified by relationships among their sides. Slopes let you test whether sides are parallel or perpendicular.

Slope tests you will use

• Parallel: two non-vertical segments are parallel if their slopes are equal. Two vertical segments are parallel to each other as well.
• Perpendicular: two non-vertical segments are perpendicular if their slopes are negative reciprocals (their product is -1). A horizontal segment is perpendicular to a vertical segment.

Which sides to compare

Label the quadrilateral in order A→B→C→D→A. Then the sides are AB, BC, CD, DA.

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• Parallelogram slope pattern: slope(AB) = slope(CD) and slope(BC) = slope(DA).
• Rectangle slope pattern: the parallelogram slope pattern plus one right angle, e.g., AB ⟂ BC (or any adjacent pair perpendicular).

Step-by-step: slope check workflow

1. Compute slopes of AB, BC, CD, DA.
2. Check whether opposite sides have equal slopes (parallel).
3. If you want a rectangle, also check whether one adjacent pair is perpendicular.

Note: Slope alone can confirm a parallelogram (both pairs of opposite sides parallel). For a rectangle, slope can confirm right angles, but you may still use distance/diagonals as an additional confirmation, especially if the point order is uncertain.

3) Use distances to test equal sides and (when needed) equal diagonals

Distance checks complement slope checks. They are especially useful when slopes are messy, when you suspect a kite/trapezoid, or when you want to confirm a rectangle by its diagonals.

Distance patterns

• Parallelogram: opposite sides have equal length: AB = CD and BC = DA.
• Rectangle: it is a parallelogram with right angles. A common coordinate-geometry confirmation is: opposite sides parallel (or equal) and diagonals equal: AC = BD.

In many problems, you can prove “rectangle” by either of these combinations:

• Method A (slope-heavy): opposite sides parallel and one right angle (perpendicular adjacent slopes).
• Method B (distance-heavy): it is a parallelogram (opposite sides parallel or equal) and diagonals are equal.

Step-by-step: distance check workflow

1. Compute side lengths AB, BC, CD, DA (often using squared distances to avoid radicals).
2. Check AB = CD and BC = DA for a parallelogram.
3. If testing for a rectangle, compute diagonals AC and BD and check whether they are equal.

Using squared distance to keep arithmetic clean

Instead of comparing distances like √29, compare squared distances like 29. Equality is preserved.

d^2((x1,y1),(x2,y2)) = (x2-x1)^2 + (y2-y1)^2

4) Practice: decide rectangle/parallelogram; find missing vertices

Practice A: Given four points, decide if they form a parallelogram or rectangle

Problem A1: Do the points A(1,1), B(5,2), C(7,6), D(3,5) (connected in order) form a parallelogram? A rectangle?

Step 1: Slopes of sides

• m_AB = (2-1)/(5-1) = 1/4
• m_BC = (6-2)/(7-5) = 4/2 = 2
• m_CD = (5-6)/(3-7) = (-1)/(-4) = 1/4
• m_DA = (1-5)/(1-3) = (-4)/(-2) = 2

Step 2: Parallel test

• m_AB = m_CD so AB ∥ CD
• m_BC = m_DA so BC ∥ DA

This confirms a parallelogram.

Step 3: Perpendicular test for rectangle

Check adjacent slopes: m_AB · m_BC = (1/4)·2 = 1/2, not -1. So it is not a rectangle.

Practice B: A tilted rectangle check using diagonals

Problem B1: Do the points A(0,1), B(4,3), C(2,7), D(-2,5) form a rectangle when connected in order?

Step 1: Check it is a parallelogram (slopes)

• m_AB = (3-1)/(4-0) = 2/4 = 1/2
• m_BC = (7-3)/(2-4) = 4/(-2) = -2
• m_CD = (5-7)/(-2-2) = (-2)/(-4) = 1/2
• m_DA = (1-5)/(0-(-2)) = (-4)/2 = -2

Opposite sides are parallel: AB ∥ CD and BC ∥ DA, so it is a parallelogram.

Step 2: Confirm rectangle (right angle via slopes)

m_AB · m_BC = (1/2)·(-2) = -1, so AB ⟂ BC. Therefore the parallelogram is a rectangle.

Optional diagonal check (distance-squared)

• AC^2 = (2-0)^2 + (7-1)^2 = 4 + 36 = 40
• BD^2 = (-2-4)^2 + (5-3)^2 = 36 + 4 = 40

Diagonals match, consistent with a rectangle.

Practice C: Find a missing vertex to complete a rectangle aligned with the grid

Problem C1: Three vertices of an axis-aligned rectangle are A(2,1), B(7,1), D(2,5), where AB is horizontal and AD is vertical. Find the missing vertex C.

Step-by-step reasoning

• Axis-aligned means sides are horizontal/vertical.
• A(2,1) to B(7,1) changes x but keeps y=1.
• A(2,1) to D(2,5) keeps x=2 but changes y.
• The missing corner C must share x with B and y with D.

So C(7,5).

Practice D: Find a missing vertex to complete a tilted rectangle (using vectors)

For a tilted rectangle, a reliable method is to use the fact that opposite sides are parallel and equal as vectors. If A is a corner with adjacent corners B and D, then the fourth vertex is:

C = B + D - A

Problem D1: Points A(1,2), B(6,4), and D(-1,7) are three vertices of a rectangle with right angle at A (so AB and AD are adjacent sides). Find C, then verify it is a rectangle.

Step 1: Compute the missing vertex

• C = B + D - A = (6 + (-1) - 1, 4 + 7 - 2) = (4,9)

So C(4,9).

Step 2: Verify right angle at A (slope or dot-product idea via slopes)

• m_AB = (4-2)/(6-1) = 2/5
• m_AD = (7-2)/(-1-1) = 5/(-2) = -5/2

m_AB · m_AD = (2/5)·(-5/2) = -1, so AB ⟂ AD.

Step 3: Verify opposite sides parallel (slopes)

• m_CD using C(4,9) and D(-1,7): (7-9)/(-1-4) = (-2)/(-5) = 2/5, matches m_AB.
• m_BC using B(6,4) and C(4,9): (9-4)/(4-6) = 5/(-2) = -5/2, matches m_AD.

This confirms a rectangle.

Practice E: Decide if four points can be rearranged to form a rectangle

Problem E1: The points are P(0,0), Q(3,1), R(4,4), S(1,3). Determine whether they can be ordered to form a rectangle.

Step 1: Try an order around the outside

A reasonable order is P→Q→R→S→P (it traces the perimeter without crossing).

Step 2: Check adjacent slopes for a right angle

• m_PQ = (1-0)/(3-0) = 1/3
• m_QR = (4-1)/(4-3) = 3

(1/3)·3 = 1, not -1, so not perpendicular. That suggests it is not a rectangle in this order.

Step 3: Use diagonal equality as a quick rectangle test candidate

For a rectangle, diagonals are equal. Compute squared diagonals for this order:

• PR^2 = (4-0)^2 + (4-0)^2 = 16+16=32
• QS^2 = (1-3)^2 + (3-1)^2 = 4+4=8

Not equal, so this ordering is not a rectangle. You can also test other non-crossing orders; if no order produces equal diagonals and right angles (or the parallelogram + equal diagonals condition), then the set cannot form a rectangle.

Checklist you can apply to any four points