NY State Math Geometry Expressing Geometric Properties with Equations. Practice it free.

Per the NYSED Geometry Educator Guide blueprint, Expressing Geometric Properties with Equations (G-GPE) is 12%–18% of credit, covering conic-section equations and coordinate proofs. This Geometry reporting domain maps to 8 practice skills and 8 representative questions from the playable bank.

GeometryG-GPE8 mapped skills
What the test measures

Expressing Geometric Properties with Equations skills

  1. Translate between the geometric description and the equation for a conic section

  2. Use coordinates to prove simple geometric theorems algebraically

Standards basis

New York State Learning Standards / New York State Next Generation Learning Standards (NGLS) — New York

How the NY State Math reports it

New York’s Grades 3–8 math tests and Regents math exams are state-developed assessments rather than a shared national framework. The grades 3–8 exams are aligned to New York State learning standards, and the Regents math sequence includes Algebra I, Geometry, and Algebra II, with the June 2026 Algebra II exam explicitly based on the New York State Next Generation Learning Standards.

Blueprint & weighting

Grades 3–8: the official page and the retrieved administration guide describe item/credit structure and timing, but no published domain-by-domain blueprint weighting was located in the primary sources used here. Regents (high school): NYSED publishes a per-conceptual-category 'percent of test by credit' blueprint for each course in its Educator Guides. Algebra I — Number & Quantity 4%–10%, Algebra 48%–61%, Functions 24%–32%, Statistics & Probability 7%–15%. Geometry (single conceptual category, by domain) — Congruence (G-CO) 27%–34%, Similarity, Right Triangles, & Trigonometry (G-SRT) 29%–37%, Circles (G-C) 2%–8%, Expressing Geometric Properties with Equations (G-GPE) 12%–18%, Geometric Measurement & Dimensions (G-GMD) 2%–8%, Modeling with Geometry (G-MG) 8%–15%. Algebra II — Number & Quantity 4%–8%, Algebra 30%–39%, Functions 38%–45%, Statistics & Probability 14%–22%.

Practice by skill

Topics mapped to this domain

Circle diameter endpoints to radius (SAT)

Grade 1010 questions

Circle equation translations (ACT/SAT/PSAT 8/9/10)

Grade 1010 questions

Possible coordinates on a circle (SAT)

Grade 1010 questions

Conic sections

Grade 1120 questions

Equation of a circle

Grade 1020 questions

Midpoint formula

Grade 1015 questions

Parallel and perpendicular lines through a point (SAT)

Grade 910 questions

Parallel and perpendicular slopes from slope-intercept form

Grade 910 questions
Try it now

8 free questions · 0/0 correct

Practice playeasy

A circle in the coordinate plane has a diameter with endpoints (2,−4)(2, -4) and (18,−4).(18, -4)\textsf{.} What is the radius of the circle?

The endpoints share the same yy-coordinate, so the diameter is horizontal with length ∣18−2∣=16.|18 - 2| = 16\textsf{.} The radius is half the diameter, so r=162=8.r = \dfrac{16}{2} = 8\textsf{.}

Practice playmedium

The equation (x−4)2+(y−2)2=36(x - 4)^2 + (y - 2)^2 = 36 represents Circle A. Circle B is obtained by shifting Circle A left 55 units in the coordinate plane. Which of the following equations represents Circle B?

Circle A has center (4,2)(4, 2) and radius 66. Shifting left 55 moves the center to (−1,2)(-1, 2), so Circle B is (x+1)2+(y−2)2=36(x + 1)^2 + (y - 2)^2 = 36.

Practice playmedium

The graph of the equation (x+8)2+(y−6)2=169(x + 8)^2 + (y - 6)^2 = 169 is a circle in the coordinate plane. The point (a,b)(a, b) lies on the circle. Which of the following is a possible value for b?b\textsf{?}

The center is (−8,6)(-8, 6) and the radius is 13.13\textsf{.} A point on the circle must satisfy 6−13≤b≤6+13,6 - 13 \le b \le 6 + 13\textsf{,} so −7≤b≤19.-7 \le b \le 19\textsf{.} The value 1111 is in this interval.

Practice playeasy

The hyperbola x216−y29=1\dfrac{x^{2}}{16} - \dfrac{y^{2}}{9} = 1 has a vertex on the positive x-axis. What is the x-coordinate of that vertex?

Set y=0y = 0: then x2=16x^{2} = 16, so the vertices are (±4,0)(\pm 4, 0) and the positive x-coordinate is 44.

Practice playeasy

A circle in the coordinate plane has equation (x−2)2+(y−7)2=36.(x - 2)^{2} + (y - 7)^{2} = 36\textsf{.} What is the measure of the diameter of the circle?

The radius measures 66 units, so the diameter measures 1212 units.

Practice playeasy

The midpoint of PQ‾\overline{PQ} is (6,2)(6, 2), and P=(9,8)P = (9, 8). What is the yy-coordinate of QQ?

Since 8+Qy2=2\dfrac{8 + Q_y}{2} = 2, we get Qy=4−8=−4Q_y = 4 - 8 = -4.

Practice playmedium

The equation of a line is y=12x−3y = \dfrac{1}{2}x - 3. Which equation represents the line that is perpendicular to this line and passes through the point (2,−1)(2, -1)?

Perpendicular lines have slopes that are negative reciprocals, so m=−2m = -2. Substitute (2,−1)(2, -1) into y=−2x+by = -2x + b: −1=−2(2)+b=−4+b-1 = -2(2) + b = -4 + b, so b=3b = 3. The equation is y=−2x+3.y = -2x + 3\textsf{.}

Practice playeasy

Line ww is defined by y=−13x+4y = -\dfrac{1}{3}x + 4. Line zz is perpendicular to line ww in the coordinate plane. What is the slope of line zz?

Perpendicular lines have slopes that are negative reciprocals. The slope of line ww is −13-\dfrac{1}{3}, so the slope of line zz is 3.3\textsf{.}

Keep practicing

Turn expressing geometric properties with equations into game time.

The NY State Math placement starts with this test's real coverage map and finds the right difficulty.

Batting Cage game coverBatting CagePlay →Blossom Leap game coverBlossom LeapPlay →Buzzer Beater game coverBuzzer BeaterPlay →Minecart Junction game coverMinecart JunctionPlay →
Related Common Core standards
HSG.GPE.A.1HSG.GPE.A.3HSG.GPE.B.6
Source and test overview
All NY State Math prepOfficial test source
Reporting categories

More Geometry domains

CongruencePer the NYSED Geometry Educator Guide blueprint, Congruence (G-CO) is 27%–34% of credit, covering transformations in the plane, congruence in terms of rigid motions, geometric theorems, and constructions.8 QsPractice →Similarity, Right Triangles, & TrigonometryPer the NYSED Geometry Educator Guide blueprint, Similarity, Right Triangles, & Trigonometry (G-SRT) is 29%–37% of credit, covering similarity transformations, theorems involving similarity, right-triangle trigonometry, and trigonometry of general triangles.8 QsPractice →CirclesPer the NYSED Geometry Educator Guide blueprint, Circles (G-C) is 2%–8% of credit, covering theorems about circles and arc lengths and areas of sectors.8 QsPractice →Geometric Measurement & DimensionsPer the NYSED Geometry Educator Guide blueprint, Geometric Measurement & Dimensions (G-GMD) is 2%–8% of credit, covering volume formulas and relationships between two- and three-dimensional objects.8 QsPractice →Modeling with GeometryPer the NYSED Geometry Educator Guide blueprint, Modeling with Geometry (G-MG) is 8%–15% of credit, applying geometric concepts in modeling situations.8 QsPractice →