Coordinate Geometry Basics: Coordinate-Based Proofs and Shape Verification

Coordinate-Based Proofs: Turning Calculations into Geometry

A coordinate-based proof is an organized argument that uses algebraic computations (like slopes, distances, and midpoints) to justify a geometric claim (like “this quadrilateral is a rectangle” or “these diagonals bisect each other”). The key skill is not the computation itself, but the reasoning link: what the computed value means and why that meaning guarantees a shape property.

A Repeatable Proof Format (Use This Template)

Use the same structure every time so your work reads like a proof rather than a list of numbers.

1. State the Given

   • List the points with labels in order (for polygons, keep a consistent order around the shape).
   • State what you are trying to prove (the claim).

2. Compute the Needed Quantities

   • Choose computations that match the claim: slopes for parallel/perpendicular, distances for congruent sides/diagonals, midpoints for bisection.
   • Show substitutions clearly so another reader can follow.

3. Interpret Each Result

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   • Translate algebra to geometry: “equal slopes → parallel,” “negative reciprocal slopes → perpendicular,” “equal distances → congruent,” “same midpoint → bisect each other.”

4. Conclude the Shape Property

   • Write a final sentence that uses the interpreted facts to justify the claim.

Common “Computation → Meaning” Links

Guided Example 1: Prove a Quadrilateral Is a Rectangle

Given: Quadrilateral ABCD with points A(1,1), B(6,3), C(4,8), D(-1,6).

Claim: ABCD is a rectangle.

Step 1: Compute slopes of consecutive sides

Compute m_AB and m_BC to check for a right angle, and compute m_AB vs. m_CD and m_BC vs. m_AD to check opposite sides parallel.

m_AB = (3 - 1) / (6 - 1) = 2/5

m_BC = (8 - 3) / (4 - 6) = 5/(-2) = -5/2

m_CD = (6 - 8) / (-1 - 4) = (-2)/(-5) = 2/5

m_AD = (6 - 1) / (-1 - 1) = 5/(-2) = -5/2

Step 2: Interpret the results

• Perpendicular sides: m_AB = 2/5 and m_BC = -5/2 are negative reciprocals, so AB ⟂ BC. That means angle ABC is a right angle.

• Opposite sides parallel: m_AB = m_CD = 2/5, so AB ∥ CD. Also m_BC = m_AD = -5/2, so BC ∥ AD.

Step 3: Conclude rectangle

Since both pairs of opposite sides are parallel, ABCD is a parallelogram. A parallelogram with one right angle is a rectangle. Therefore, ABCD is a rectangle.

Optional Verification (Diagonal Check)

Some proofs also confirm that diagonals are congruent (a rectangle property). You could compute AC and BD and show they are equal. This is not necessary here because the right-angle-in-a-parallelogram argument already completes the proof.

Guided Example 2: Prove Diagonals Bisect Each Other in a Parallelogram (Using Midpoints)

Given: Parallelogram PQRS with points P(2,1), Q(8,3), R(10,9), S(4,7).

Claim: Diagonals PR and QS bisect each other.

Step 1: Compute the midpoint of each diagonal

Find midpoint of PR:

M_PR = ((2 + 10)/2, (1 + 9)/2) = (12/2, 10/2) = (6, 5)

Find midpoint of QS:

M_QS = ((8 + 4)/2, (3 + 7)/2) = (12/2, 10/2) = (6, 5)

Step 2: Interpret the results

• The diagonals PR and QS have the same midpoint, (6,5).
• If two segments share a midpoint, then each segment is cut into two equal parts at that point. That is exactly what it means to bisect.

Step 3: Conclude bisection

Because M_PR = M_QS, diagonals PR and QS bisect each other.

Note on Proof Choices

This midpoint method is a direct coordinate translation of the geometric statement “diagonals of a parallelogram bisect each other.” It avoids extra steps like finding side lengths or angles because the claim is specifically about bisection.

Practice: Write Explanations That Connect Algebra to Geometry

For each prompt, your work must include both (a) computations and (b) sentences that interpret them. Use the proof format template.

Practice Set A: Identify the Best Computations

• A1. You want to prove a quadrilateral is a rectangle. Which computations are most efficient: slopes, distances, midpoints, or a combination? Write a 2–3 sentence plan (no calculations yet).

• A2. You want to prove two diagonals bisect each other. What single type of computation is most direct? Explain why in one sentence.

• A3. You computed two slopes and got 3/4 and 3/4. Write a sentence that states the geometric conclusion.

• A4. You computed two slopes and got -2 and 1/2. Write a sentence that states the geometric conclusion.

• A5. You computed two midpoints and both were (-1, 6). Write a sentence that states the geometric conclusion about the segments.

Practice Set B: Short Coordinate Proofs (Write in Full Sentences)

For each problem, show computations and then write a short paragraph (3–6 sentences) that links results to the claim.

• B1. Prove a rectangle. Points A(-2,0), B(3,2), C(1,7), D(-4,5). Prove ABCD is a rectangle.

• B2. Prove diagonals bisect. Points P(-1,-2), Q(5,0), R(7,6), S(1,4). Prove diagonals PR and QS bisect each other.

• B3. Decide what you can prove. Points J(0,0), K(4,1), L(6,5), M(2,4). Compute enough to determine whether JKLM is a parallelogram, a rectangle, both, or neither. Your written conclusion must name the classification you can justify from your computations.

Final Assessment: Multi-Step Verification Tasks

For each task, you must: (1) make a quick sketch or plot (on graph paper or a coordinate grid), (2) show the needed computations, and (3) write a clear conclusion that states the classification and cites the properties you verified.

Assessment 1: Verify a Rectangle from Coordinates

Given: A(0,2), B(6,5), C(3,11), D(-3,8)

• Plot the points in order A → B → C → D → A.
• Compute slopes to verify both pairs of opposite sides are parallel.
• Compute slopes of one pair of adjacent sides to verify a right angle.
• Write a conclusion that uses the words parallel, perpendicular, and rectangle correctly.

Assessment 2: Verify a Parallelogram via Diagonal Bisection

Given: P(1,1), Q(9,3), R(11,10), S(3,8)

• Plot the points in order P → Q → R → S → P.
• Compute the midpoint of diagonal PR.
• Compute the midpoint of diagonal QS.
• Write a conclusion explaining why equal midpoints imply the diagonals bisect each other, and state what that tells you about the quadrilateral.

Assessment 3: Mixed Evidence Classification

Given: W(-2,3), X(4,6), Y(7,0), Z(1,-3)

• Plot the points in order.
• Compute slopes of all four sides.
• Compute midpoints of both diagonals.
• Based on your results, classify the quadrilateral as precisely as you can (for example: parallelogram, rectangle, or neither). Your written conclusion must reference at least two computed facts and their geometric meanings.