How do I earn all 6 credits on the Part IV Regents Geometry question?

Each sub-part is graded separately. On Q35 January 2026 you must (a) prove ABCD is a trapezoid by showing exactly one pair of opposite sides has equal slopes, (b) show slope of EF equals the slope of AD and BC to prove EF is parallel to both, and (c) use the distance formula to compute AD, BC, and EF, then check whether EF equals ½(AD + BC). Every calculation must appear on paper.

How do I prove a quadrilateral is a trapezoid on the coordinate plane?

Compute the slopes of all four sides using the slope formula. Show that exactly one pair of opposite sides has equal slopes (so that pair is parallel) and that the other pair does not (so those two sides are not parallel). By the definition of a trapezoid — a quadrilateral with exactly one pair of parallel sides — the quadrilateral is a trapezoid. Some Regents textbooks use the inclusive definition (at least one pair), in which case showing one pair is enough.

How do I prove two coordinate segments are parallel?

Use the slope formula on each segment. If the two slopes are equal, the segments are parallel. On Q35, slope EF = (5 − (−3)) / (3 − (−3)) = 8/6 = 4/3, and slope AD = slope BC = 4/3, so EF is parallel to both AD and BC.

← SOMATH Blog · Regents Prep NYC · Updated August 4, 2026

Regents Geometry January 2026 — Part IV Answer & Full Solution

Free worked solution for the 6-credit Part IV question on the January 2026 New York State Regents Geometry exam. This is the longest question on the test — a coordinate-geometry problem with four sub-parts. This is the final post in our four-post series covering all 35 questions.

📄 Original NYSED exam (PDF)

All diagrams, coordinate grids, and reference sheet are in the official New York State Education Department release. Open it in a second tab so you can see the figure as you work through the question below.

Download the January 2026 Geometry Regents PDF

How Part IV is graded. Part IV is the single hardest question on the test, worth 6 credits. A correct numerical answer with no work receives only 1 credit. Each of the three sub-parts is graded separately, so partial credit is available — do not skip a sub-part just because you are stuck on an earlier one. Use the coordinate grid printed with the question to visualize the points, but every algebraic step must still be written out.

What Part IV tests

Every recent Geometry Regents ends with a large coordinate-geometry question. The Regents wants to know whether you can move fluently between four core coordinate tools — slope, midpoint, distance, and the equation of a line — and whether you can attach the right theorem to each calculation. This January 2026 question stitches together all three foundational coordinate formulas and then asks you to reach a shape classification (trapezoid), a parallel-lines conclusion (segment EF parallel to two sides), and a length verification for the trapezoid midsegment theorem.

The three coordinate formulas you will use

Slope from (x₁, y₁) to (x₂, y₂): m = (y₂ − y₁) / (x₂ − x₁).
Midpoint: M = ((x₁ + x₂) / 2, (y₁ + y₂) / 2).
Distance: d = √((x₂ − x₁)² + (y₂ − y₁)²).

How to prove a quadrilateral is a trapezoid (coordinate plane)

Compute the slopes of all four sides.
Show exactly one pair of opposite sides has equal slopes (that pair is parallel).
Show the other pair does not have equal slopes (that pair is not parallel).
State the conclusion: by the definition of a trapezoid, the quadrilateral is a trapezoid.

Parallel lines from equal slopes

Two non-vertical lines are parallel if and only if they have the same slope. Two vertical lines (undefined slope) are also parallel to each other, but a vertical line is not parallel to a line of slope 0 — those two are perpendicular. Watch for vertical sides on the coordinate plane; they need a sentence noting the undefined slope, not a numerical value.

Trapezoid midsegment theorem

The midsegment of a trapezoid — the segment connecting the midpoints of the two non-parallel sides (the legs) — is parallel to both bases and equal in length to half the sum of the two bases. Symbolically: if the bases are b₁ and b₂, the midsegment has length ½(b₁ + b₂). This is the theorem being verified in part (c) of Q35.

Part IV — Question 35 (6 credits)

Coordinate geometry · four sub-parts · click the card to reveal the full worked solution.

Question 35

Quadrilateral ABCD is graphed on the set of axes with vertices at A(−3, 1), B(−3, −7), C(6, 5), and D(0, 5). Segment EF is graphed with endpoints E(−3, −3) and F(3, 5).

(a) Prove ABCD is a trapezoid.
(b) Use coordinate geometry to prove EF is parallel to AD and BC.
(c) Is EF = ½(AD + BC)? Use coordinate geometry to justify your answer.

Part (a)

Prove ABCD is a trapezoid

Method. A trapezoid is a quadrilateral with exactly one pair of parallel opposite sides. Compute all four side slopes and show only one pair is equal.

Slope of AB from A(−3, 1) to B(−3, −7):

The x-coordinates are equal, so AB is a vertical segment. Its slope is undefined.

Slope of CD from C(6, 5) to D(0, 5):

m_CD = (5 − 5) / (0 − 6) = 0 / (−6) = 0. CD is horizontal.

AB and CD are not parallel (undefined slope vs. 0 — a vertical line and a horizontal line are perpendicular, not parallel).

Slope of AD from A(−3, 1) to D(0, 5):

m_AD = (5 − 1) / (0 − (−3)) = 4 / 3.

Slope of BC from B(−3, −7) to C(6, 5):

m_BC = (5 − (−7)) / (6 − (−3)) = 12 / 9 = 4 / 3.

m_AD = m_BC = 4/3, so AD ∥ BC.

Exactly one pair of opposite sides is parallel (AD ∥ BC), so ABCD is a trapezoid (definition of a trapezoid). ■

Rubric. Full credit for (a) requires (i) both slope calculations for the parallel pair, (ii) evidence that the other pair is not parallel (state the undefined slope on AB and the 0 slope on CD, or explicitly say a vertical line and a horizontal line are not parallel), and (iii) the concluding statement citing the definition of a trapezoid. Skipping the “not parallel” step on AB and CD is the most common credit loss on this sub-part.

Part (b)

Prove EF ∥ AD and EF ∥ BC

Method. Compute the slope of EF and compare it to the slopes of AD and BC from part (a).

Slope of EF from E(−3, −3) to F(3, 5):

m_EF = (5 − (−3)) / (3 − (−3)) = 8 / 6 = 4 / 3.

From part (a), m_AD = 4/3 and m_BC = 4/3.

∴ m_EF = m_AD = m_BC = 4/3, so EF ∥ AD and EF ∥ BC (two non-vertical lines are parallel if and only if they have the same slope). ■

Rubric. Full credit for (b) requires the slope formula on EF with substituted numbers, an explicit comparison to both AD and BC, and a concluding sentence citing the “same slope implies parallel” theorem. If part (a) is skipped, the slopes of AD and BC must be recomputed here.

Part (c)

Is EF = ½(AD + BC)?

Method. Use the distance formula to compute the exact lengths of AD, BC, and EF, then compare EF to ½(AD + BC).

Length of AD from A(−3, 1) to D(0, 5):

AD = √((0 − (−3))² + (5 − 1)²) = √(3² + 4²) = √(9 + 16) = √25 = 5.

Length of BC from B(−3, −7) to C(6, 5):

BC = √((6 − (−3))² + (5 − (−7))²) = √(9² + 12²) = √(81 + 144) = √225 = 15.

Length of EF from E(−3, −3) to F(3, 5):

EF = √((3 − (−3))² + (5 − (−3))²) = √(6² + 8²) = √(36 + 64) = √100 = 10.

Test the identity: ½(AD + BC) = ½(5 + 15) = ½(20) = 10.

EF = 10 and ½(AD + BC) = 10, so the two are equal.

Yes, EF = ½(AD + BC). ■

Rubric. Full credit for (c) requires (i) the distance formula on all three segments with substituted numbers, (ii) the arithmetic ½(5 + 15) = 10 shown, and (iii) an explicit “yes” answer. Stating the numeric distances without the distance-formula work drops credit — the problem says “use coordinate geometry.”

Why this works — the midsegment theorem. E(−3, −3) is the midpoint of leg AB (halfway between (−3, 1) and (−3, −7)) and F(3, 5) is the midpoint of leg DC (halfway between (0, 5) and (6, 5)). So segment EF is the midsegment of trapezoid ABCD. The trapezoid midsegment theorem says the midsegment is parallel to both bases and equal in length to half their sum — exactly what parts (b) and (c) verified with coordinates.

Theory. The three coordinate tools have three matching purposes: slope answers “are these lines parallel or perpendicular?”, midpoint answers “what is the point exactly between these two?”, and distance answers “how long is this segment?” Every coordinate-geometry Regents Part IV question can be dissected into applications of one or more of these three formulas.

Big ideas to remember on test day

Attack the sub-parts in any order. Each of the three sub-parts is graded separately. If you get stuck on the trapezoid proof, jump to the distance-formula sub-part and come back.
Trapezoid = exactly one pair of parallel sides. You must show the “other” pair is not parallel, not just that one pair is.
Vertical vs. horizontal is not parallel. A vertical segment has undefined slope, a horizontal one has slope 0 — they are perpendicular. Do not skip this observation on part (a).
Same slope means parallel. That is the whole content of part (b): compute slope of EF, match it to slopes of AD and BC, cite the theorem.
Distance formula = three lines of work. Formula, substituted, simplified. Skip any of the three and you may lose a credit.
Leave the radical only when it is a radical. On Q35 all three distances simplify to whole numbers (5, 15, 10) — that is not always the case, and on other exams √13 or √50 must be left in exact form.
Write the concluding sentence. “∴ ABCD is a trapezoid because exactly one pair of opposite sides is parallel.” “∴ EF ∥ AD and EF ∥ BC.” “Yes, EF = ½(AD + BC).” These sentences are how the grader knows you finished the proof.
Recognize the midsegment. Once you see that E is the midpoint of AB and F is the midpoint of DC, part (c) is guaranteed to work out by the midsegment theorem — the coordinate work is just verifying the theorem.

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This walkthrough is provided for educational purposes. The January 2026 Geometry Regents exam is publicly released by the New York State Education Department. Questions and figures are the property of NYSED.

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Regents Geometry January 2026 — Part IV Answer & Full Solution

📄 Original NYSED exam (PDF)

Sections

What Part IV tests

The three coordinate formulas you will use

How to prove a quadrilateral is a trapezoid (coordinate plane)

Parallel lines from equal slopes

Trapezoid midsegment theorem

Part IV — Question 35 (6 credits)

Prove ABCD is a trapezoid

Prove EF ∥ AD and EF ∥ BC

Is EF = ½(AD + BC)?

Big ideas to remember on test day

More Regents Geometry help

About SOMATH

Want your child in a Regents Geometry & Trigonometry Prep class at SOMATH?