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Geometry · Recognition Drill

Geometry · 30 stems · 8s per card
30 contest stems on plane and solid geometry. Classify each — don't compute. 5-second budget (8 in coach mode).
Year-8 notation glossary — read this first
∠ABC\angle ABC
Angle at vertex BB formed by rays BABA and BCBC.
△ABC\triangle ABC
Triangle with vertices A,B,CA, B, C.
AB‾\overline{AB} / ABAB
Segment from AA to BB, or its length depending on context.
∥\parallel / ⊥\perp
Parallel / perpendicular.
sin⁡θ,cos⁡θ,tan⁡θ\sin \theta, \cos \theta, \tan \theta
Trig ratios. In a right triangle: opp/hyp, adj/hyp, opp/adj.
π\pi (pi)
Ratio of a circle's circumference to its diameter ≈ 3.141593.14159.
(h,k),r(h, k), r
Centre (h,k)(h, k) and radius rr of a circle.
u,v\mathbf u, \mathbf v (bold)
Vectors — quantities with magnitude AND direction. In 2D: u=(u1,u2)\mathbf u = (u_1, u_2).
Technique deck — what the button labels mean
Pythagorean theorem
$a^2 + b^2 = c^2$ for the legs and hypotenuse of a right triangle.
Heron's formula
Area of a triangle from sides: $\sqrt{s(s-a)(s-b)(s-c)}$, $s = (a+b+c)/2$.
Shoelace formula
Area of a polygon from vertex coordinates: $\tfrac12 |\sum (x_i y_{i+1} - x_{i+1} y_i)|$.
Power of a point
$PA \cdot PB = PC \cdot PD$ for two chords through $P$; tangent: $PT^2 = PA \cdot PB$.
Similar triangles
AA, SAS, SSS similarity. Ratios: sides scale by $k$, area by $k^2$, volume by $k^3$.
Inscribed angle / central angle
Inscribed angle = $\tfrac12$ central angle on the same arc; semicircle ⇒ $90°$.
Cyclic quadrilateral
Opposite angles sum to $180°$; converse also true. Use Ptolemy for products.
Ptolemy's theorem
For cyclic $ABCD$: $AC \cdot BD = AB \cdot CD + AD \cdot BC$.
Tangent-chord (alternate segment)
Angle between tangent and chord = inscribed angle in the alternate segment.
Angle chasing
Track angle equalities (parallel lines, isoceles, inscribed) until the target falls out.
Law of cosines
$c^2 = a^2 + b^2 - 2ab\cos C$ — for a triangle with one known angle.
Law of sines
$\tfrac{a}{\sin A} = \tfrac{b}{\sin B} = \tfrac{c}{\sin C} = 2R$.
SOH-CAH-TOA
$\sin = $ opp/hyp, $\cos = $ adj/hyp, $\tan = $ opp/adj — in a right triangle.
Trig identity (Pythagorean)
$\sin^2 \theta + \cos^2 \theta = 1$.
Sum-to-product
$\sin A + \sin B = 2 \sin \tfrac{A+B}{2} \cos \tfrac{A-B}{2}$ (and similar).
Centroid
Average of vertices: $G = (A+B+C)/3$; divides each median $2{:}1$.
Incenter / circumcenter
Angle bisectors → $I$ (incircle center); $\perp$ bisectors → $O$ (circumcircle center).
Orthocenter / Euler line
Altitudes meet at $H$; $O$, $G$, $H$ collinear with $\vec{OG} : \vec{GH} = 1 : 2$.
Stewart's theorem
Cevian length from sides: $b^2 m + c^2 n - a d^2 = a m n$.
Median to hypotenuse
In a right triangle, median to the hypotenuse equals half the hypotenuse.
Angle bisector theorem
Bisector from $A$ divides $BC$ so that $BD : DC = AB : AC$.
Distance formula
$\sqrt{(\Delta x)^2 + (\Delta y)^2}$.
Slope / perpendicular slopes
Slope $= \Delta y / \Delta x$; perpendicular lines have $m_1 m_2 = -1$.
Equation of a circle
$(x - h)^2 + (y - k)^2 = r^2$ — centre $(h, k)$, radius $r$.
Point-to-line distance
$|ax_0 + by_0 + c| / \sqrt{a^2 + b^2}$.
Rotation / reflection / translation
Rigid motions preserving distance. $90°$ CCW: $(x, y) \to (-y, x)$.
Volume formulas
Prism $Bh$, pyramid/cone $\tfrac13 Bh$, sphere $\tfrac{4}{3}\pi r^3$.
Surface area formulas
Sphere $4\pi r^2$; cylinder $2\pi r h + 2\pi r^2$.
Conic identification
Discriminant of $Ax^2 + Bxy + Cy^2$: parabola if $B^2 - 4AC = 0$, ellipse $< 0$, hyperbola $> 0$.
Dot product
$\mathbf u \cdot \mathbf v = u_1 v_1 + u_2 v_2$; perpendicular ⇔ dot product zero.