GRE Geometry: Triangles, Circles, Coordinate Geometry, and “Hidden” Constraints

The GRE is now a shorter, section‑adaptive test, but the Quant content and question types have not changed. You’ll still see Quantitative Comparison, single‑answer multiple choice, multiple‑answer (select all that apply), Numeric Entry, and Data Interpretation. For Quant specifically: two sections totaling 27 questions are allotted 47 minutes (Section 1: 12 questions in 21 minutes; Section 2: 15 questions in 26 minutes). An on‑screen calculator is available. Figures in pure geometry questions are not drawn to scale; coordinate axes and data graphics are drawn to scale. Net takeaway: the geometry you study is the same, but you have fewer total questions and similar time per question—so efficiency matters more than ever.

Most geometry misses come from computing before constraining. Train yourself to do this in the first 20–30 seconds: 1) Sketch a clean diagram if none is given. 2) Underline or rewrite the givens as marks on the figure: equal sides/angles, parallels, midpoints, perpendiculars, tangents, diameters. 3) Add “must‑exist” lines: radii to points of tangency, the center to a chord’s midpoint, altitudes in right or isosceles triangles. 4) Decide the win condition: are you proving equality/ratio (favor similar triangles and angle chasing) or finding a number (favor one clean variable and ratios to avoid messy arithmetic). When time is tight, constrain first; compute only what the question actually asks.

Before touching numbers, mark: 1) Side/angle equalities. In isosceles Δs, base angles are equal; the median to the base is also the altitude and angle bisector. 2) Right‑triangle structure. If you see a right angle, list the handy triples (3‑4‑5, 5‑12‑13, 8‑15‑17, 7‑24‑25) and the special ratios: 45‑45‑90 has x, x, x√2; 30‑60‑90 has x, x√3, 2x. 3) Shared altitudes/heights. If two triangles share a height, their area ratio equals the ratio of their bases, and vice versa. 4) Similarity triggers. Parallels, angle bisection, or an altitude to the hypotenuse in a right triangle often create pairs of similar triangles. Similarity is your speed tool for ratios and lengths without solving full systems.

1. Midpoint connector (triangle midsegment): the segment joining midpoints of two sides is parallel to the third side and half its length. 2) Median to hypotenuse in a right triangle equals half the hypotenuse. 3) Triangle inequality for sanity checks: the sum of any two sides must exceed the third; use it to eliminate impossible answer choices fast. 4) Area with no height given: rotate perspective—choose a different base where the height is obvious, or place the triangle on coordinate axes for a quick 1/2·base·height count.

1. Draw radii to any point mentioned on the circle; connect the center to chord midpoints. 2) If a tangent appears, draw the radius to the point of tangency and mark the right angle. 3) If a diameter is mentioned or you can draw one, remember an angle subtending a diameter is 90°. 4) Equal chords subtend equal arcs and sit the same distance from the center. 5) For arc/sector questions, keep everything in ratios: arc length or sector area is proportional to central angle; working in ratios avoids π and r until the end.

1. Tangent‑chord right angle: the angle between a tangent and a radius is 90°, creating instant right triangles. 2) Inscribed‑central link: an inscribed angle equals half its intercepted central angle—convert inscribed angles to central angles when arc measures are easier to handle. 3) Perpendicular bisector rule: the line through a chord’s midpoint and the center is perpendicular to the chord—use it to locate centers or radii quickly. 4) Two tangents from the same external point are equal in length; the triangle formed with the center is often isosceles, so equal angles follow.

1. Slope‑first thinking: parallel lines share slope; perpendicular slopes are negative reciprocals. 2) Intercepts are free points—use them as triangle bases/heights to get area quickly. 3) Compare lengths by squared distance to avoid square roots. 4) The circle equation (x−h)²+(y−k)²=r² helps only if a center and one point are clean; otherwise favor right‑triangle distances and perpendiculars. 5) Midpoint and perpendicular bisector logic let you find unknown centers or check equality conditions with minimal algebra.

In QC, you win by building or breaking a claim quickly. Try: 1) A simple legal case. Construct the most symmetric figure that satisfies the givens (isosceles, right, or center‑aligned) and compare the quantities. 2) A different legal case. Nudge one length or angle to an extreme while keeping the constraints. If the relationship flips, the answer is “cannot be determined.” 3) Ratio thinking beats arithmetic. If both quantities scale together (e.g., similar triangles), equality can persist across many valid cases.

1. Triangles (isosceles). Given AB=AC and D is the midpoint of BC. What’s special about AD? Constraint marks show AD is simultaneously a median, altitude, and angle bisector—three properties for the price of one. 2) Circles (tangent). Line PT touches circle O at T; OT=5 and PT=12. Draw OT ⟂ PT, then OP is the hypotenuse of a 5‑12‑13 triangle, so OP=13 without extra computation. 3) Coordinates (perpendiculars). A line through (0,3) is perpendicular to a line of slope −2/3. Its slope is 3/2; set y−3=(3/2)(x−0). The x‑intercept comes from y=0: x=−2—no long algebra. 4) QC (regular hexagon in a circle). Quantity A: side of the hexagon. Quantity B: circle radius. Mark the 60° central angles; a side is a chord subtending 60°, so side length equals the radius; answer is equality.

Use these time rails. 0–20 seconds: draw and mark constraints. 20–60 seconds: pick a path—similarity/ratios for comparisons, one clean variable for numeric results. If you lack a clear relation by 60–75 seconds, flag and move. When returning, swap methods (e.g., place on axes, use area ratios instead of lengths). Keep arithmetic lazy: cancel common factors before multiplying, compare squared lengths, and postpone π until the final step.

Triangles: area = 1/2·b·h; 45‑45‑90 has legs x and hypotenuse x√2; 30‑60‑90 has short leg x, long leg x√3, hypotenuse 2x; Pythagorean triples: 3‑4‑5, 5‑12‑13, 8‑15‑17, 7‑24‑25 and their multiples. Circles: circumference 2πr; area πr²; arc length and sector area are proportional to the central angle. Coordinates: slope (y2−y1)/(x2−x1); distance²=(Δx)²+(Δy)²; midpoint is the average of coordinates. Learn these cold; everything else should flow from constraints and similarity.

1. Trusting the picture. Unless it’s a coordinate graph or a data display, do not measure; add marks and reason. 2) Solving for what you don’t need. If the question asks for a ratio, stay in ratios; if it asks which is greater, don’t compute exact values. 3) Forgetting implied right angles. Tangents and altitudes create 90° automatically—draw and mark them. 4) Parallel lines left unlabeled. Mark corresponding/alternate interior angles so that similarity pops. 5) Unit drift. Radii in centimeters and arcs in degrees? Keep units and angle modes consistent.

Start with a short diagnostic session focused on geometry to see your current accuracy and pacing on triangles, circles, and coordinate geometry. Then use the Learn → Solve Together → Test Yourself flow: learn constraint‑first tactics in concise lessons; in Solve Together, walk through real GRE‑style items with layered hints that prompt you to add the right marks rather than grind arithmetic; finally, run 10–15 minute Test Yourself sets that mimic the 1:45‑per‑question feel. As you improve, Exambank auto‑adjusts difficulty, serves targeted review of your weak subskills (e.g., tangent geometry or slope/perpendicular basics), and tracks your time per item so you know when to skip and return.

Did I 1) draw radii to tangents and mark 90°; 2) use diameter ⇒ right angle if applicable; 3) mark all equal sides/angles; 4) exploit parallels for similar triangles; 5) compare with ratios instead of raw numbers; 6) keep everything in squared lengths when comparing distances; 7) choose the friendliest base/height; 8) check that my result satisfies triangle inequality; 9) confirm units; 10) answer exactly what was asked.

On the shorter GRE, geometry rewards test‑takers who constrain first and compute second. If you habitually draw the must‑have lines, label the must‑be relations, and choose ratio‑based paths, you’ll cut time, avoid traps, and convert hard‑looking figures into routine wins.