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

Each sub-part must be shown separately. You must (a) state the coordinates of D and show the reasoning, (b) prove ABCD is a parallelogram with a valid method (opposite sides parallel using slopes is standard), (c) state the midpoint E of BC using the midpoint formula, and (d) prove △ABE is isosceles by showing two sides are congruent with the distance formula. Every calculation must appear on paper.

How do I find the missing fourth vertex of a parallelogram from three given vertices?

In parallelogram ABCD, the diagonals AC and BD bisect each other. Set the midpoint of AC equal to the midpoint of BD and solve for D. For A(-3,-1) and C(-1,8), midpoint of AC is (-2, 3.5). Setting (-5+Dx)/2 = -2 gives Dx = 1, and (2+Dy)/2 = 3.5 gives Dy = 5. So D(1, 5).

What is the easiest way to prove a quadrilateral on the coordinate plane is a parallelogram?

Show that both pairs of opposite sides have equal slopes, so both pairs are parallel. For ABCD with A(-3,-1), B(-5,2), C(-1,8), D(1,5): slope AB = -3/2, slope DC = -3/2, slope BC = 3/2, slope AD = 3/2. Both pairs are parallel, so ABCD is a parallelogram.

How do I prove a coordinate-geometry triangle is isosceles?

Use the distance formula to compute the lengths of two sides. If they are equal, the triangle is isosceles (a triangle with at least two congruent sides). Show every distance-formula substitution on paper.

← SOMATH Blog · Regents Prep NYC · Updated July 28, 2026

Regents Geometry June 2026 — Part IV Answer & Full Solution

Free worked solution for the 6-credit Part IV question on the June 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 June 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 four 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 optional grid on the following page 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 June 2026 question stitches together all three foundational coordinate formulas and then asks you to reach a shape classification (parallelogram) and a triangle classification (isosceles).

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).

Ways to prove a quadrilateral is a parallelogram (coordinate plane)

Show both pairs of opposite sides are parallel (equal slopes). ← used here.
Show both pairs of opposite sides are congruent (equal distances).
Show one pair of opposite sides is both parallel and congruent.
Show the diagonals bisect each other (equal midpoints).

Isosceles triangle definition

A triangle with at least two congruent sides is isosceles. On the coordinate plane, prove isosceles by computing two side lengths with the distance formula and showing they are equal.

Part IV — Question 35 (6 credits)

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

Question 35

Triangle ABC has vertices A(−3, −1), B(−5, 2), and C(−1, 8).

(a) State the coordinates of point D such that quadrilateral ABCD is a parallelogram.
(b) Prove ABCD is a parallelogram.
(c) State the coordinates of point E, the midpoint of BC.
(d) Prove ▵ABE is an isosceles triangle.

Part (a)

Find the fourth vertex D

Idea. In parallelogram ABCD, the diagonals are AC and BD, and they bisect each other. That means the midpoint of AC equals the midpoint of BD.

Midpoint of AC: ((−3 + (−1))/2, (−1 + 8)/2) = (−4/2, 7/2) = (−2, 3.5).

Set midpoint of BD equal: ((−5 + D_x) / 2, (2 + D_y) / 2) = (−2, 3.5).

Solve each coordinate:

(−5 + D_x) / 2 = −2 ⇒ −5 + D_x = −4 ⇒ D_x = 1.
(2 + D_y) / 2 = 3.5 ⇒ 2 + D_y = 7 ⇒ D_y = 5.

D = (1, 5)

Part (b)

Prove ABCD is a parallelogram (slopes method)

Method. If both pairs of opposite sides have equal slopes, both pairs are parallel, so ABCD is a parallelogram.

Slope of AB from A(−3, −1) to B(−5, 2):

m_AB = (2 − (−1)) / (−5 − (−3)) = 3 / (−2) = −3/2.

Slope of DC from D(1, 5) to C(−1, 8):

m_DC = (8 − 5) / (−1 − 1) = 3 / (−2) = −3/2.

m_AB = m_DC, so AB ∥ DC.

Slope of BC from B(−5, 2) to C(−1, 8):

m_BC = (8 − 2) / (−1 − (−5)) = 6 / 4 = 3/2.

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

m_AD = (5 − (−1)) / (1 − (−3)) = 6 / 4 = 3/2.

m_BC = m_AD, so BC ∥ AD.

Both pairs of opposite sides are parallel, so ABCD is a parallelogram (definition of parallelogram).

Rubric. Full credit here requires all four slope calculations shown, both parallel-side conclusions stated, and the parallelogram conclusion invoking the definition (or the “both pairs parallel” theorem). Skipping the concluding statement about parallel sides loses a credit even when the four slopes are correct.

Part (c)

Find the midpoint E of BC

E = ((x_B + x_C) / 2, (y_B + y_C) / 2) = ((−5 + (−1)) / 2, (2 + 8) / 2) = (−6/2, 10/2).

E = (−3, 5)

Part (d)

Prove ▵ABE is isosceles

Method. Compute the three side lengths using the distance formula and show at least two of them are equal.

Length of AB from A(−3, −1) to B(−5, 2):

AB = √((−5 − (−3))² + (2 − (−1))²) = √((−2)² + 3²) = √(4 + 9) = √13.

Length of BE from B(−5, 2) to E(−3, 5):

BE = √((−3 − (−5))² + (5 − 2)²) = √(2² + 3²) = √(4 + 9) = √13.

AB = BE = √13, so ▵ABE has two congruent sides.

∴ ▵ABE is isosceles (a triangle with at least two congruent sides is isosceles). ■

Rubric. Full credit for (d) requires (i) using the distance formula on paper — do not just state the numeric distances, (ii) showing at least two sides are equal, and (iii) invoking the definition of isosceles. Computing all three sides is optional and does not add credit, but many students prefer it as a check.

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. If you are stuck on the parallelogram proof, do parts (c) and (d) first — each sub-part is graded separately.
Diagonals bisect each other is the fastest route to finding a missing parallelogram vertex from three given ones. Midpoint = midpoint.
Four slopes for a parallelogram, three sides for an isosceles. Match the tool to the goal.
Show every substitution. The distance formula counts as three lines of work: formula, substituted, simplified. Skip any of the three and you may lose a credit.
Leave the radical. √13 is exact; do not decimal-approximate unless the problem asks for it.
Write the concluding sentence. “∴ ABCD is a parallelogram because both pairs of opposite sides are parallel.” “∴ ▵ABE is isosceles because AB = BE.” These sentences are how the grader knows you finished the proof.

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This walkthrough is provided for educational purposes. The June 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 June 2026 — Part IV Answer & Full Solution

📄 Original NYSED exam (PDF)

Sections

What Part IV tests

The three coordinate formulas you will use

Ways to prove a quadrilateral is a parallelogram (coordinate plane)

Isosceles triangle definition

Part IV — Question 35 (6 credits)

Find the fourth vertex D

Prove ABCD is a parallelogram (slopes method)

Find the midpoint E of BC

Prove ▵ABE is isosceles

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?