Mathematical Facts, Constants & Formulas
Grade 10: Deductive Proofs, Trigonometry (SOH-CAH-TOA) & Circles
~10 min read
HESTEN ACADEMIC REFERENCE ARCHIVE
CALL NO: QA453.H47 2026 • DEWEY: 516.3 • VOL. X
Volume X • Euclidean Geometry & Trigonometry Codex
Tenth Grade Mathematics Reference Codex (High School Geometry)
The Complete Scholastic Guide to Deductive Proofs, Triangle Congruence & Similarity, SOH-CAH-TOA, Special Triangles, Circle Theorems, Arc & Sector Mensuration & Synoptic Concordance
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An exhaustive reference codex compiled for secondary geometry scholars, honors students, and collegiate preparatory researchers. Codifying Euclidean deductive proof structures, rigorous axiomatic congruence and similarity criteria, right triangle trigonometry, coordinate conic circles, differential arc measures, and computational geometric heuristics.
§ 10.1
Formal Logic, Deductive Proofs & CPCTC
Conditional & Variations
$$\begin{aligned} \text{Conditional: } &p \to q \\ \text{Converse: } &q \to p \\ \text{Inverse: } &\sim p \to \sim q \\ \text{Contrapositive: } &\sim q \to \sim p \ (\equiv p \to q) \end{aligned}$$
A conditional statement and its contrapositive are logically equivalent!
The CPCTC Axiom
$$\Delta ABC \cong \Delta DEF \implies \begin{cases} \overline{AB}\cong\overline{DE}, \overline{BC}\cong\overline{EF}, \overline{AC}\cong\overline{DF} \\ \angle A\cong\angle D, \angle B\cong\angle E, \angle C\cong\angle F \end{cases}$$
Corresponding Parts of Congruent Triangles are Congruent. Proves segment/angle equality after establishing triangle congruence.
§ 10.2
Triangle Congruence Postulates & Theorems ($\cong$)
| Congruence Criterion | Required Geometric Evidence | Scholastic Invariant Note |
|---|---|---|
| SSS (Side-Side-Side) | All 3 pairs of corresponding sides are congruent ($\cong$) | Rigidly locks all 3 interior angles |
| SAS (Side-Angle-Side) | 2 sides and the included angle between them | Angle MUST lie strictly between the two sides |
| ASA (Angle-Side-Angle) | 2 angles and the included side between them | Side connects the two angle vertices |
| AAS (Angle-Angle-Side) | 2 consecutive angles and a non-included side | Equivalent to ASA by Triangle Angle Sum ($180^\circ$) |
| HL (Hypotenuse-Leg) | Right triangle: Congruent hypotenuse and 1 leg | Valid strictly in right triangles ($90^\circ$) |
| AAA & SSA | INVALID FOR CONGRUENCE! AAA proves similarity only. SSA produces the ambiguous two-triangle case. | |
§ 10.3
Triangle Similarity ($\sim$) & Geometric Mean Theorems
THEOREM 10.3.1
Topic: Geometric Mean Altitude & Leg Theorems
In right triangle $\Delta ABC$ with altitude $h$ drawn to hypotenuse $c$, partitioning $c$ into segments $p$ and $q$:
$$h^2 = p \cdot q \implies h = \sqrt{p \cdot q} \quad (\text{Altitude is Geometric Mean of Segments})$$
$$a^2 = p \cdot c \implies a = \sqrt{p \cdot c} \quad \text{and} \quad b^2 = q \cdot c \implies b = \sqrt{q \cdot c}$$
§ 10.4
Right Triangle Trigonometry: SOH-CAH-TOA
Sine Ratio ($\sin\theta$)
$$\sin\theta = \frac{\text{Opposite}}{\text{Hypotenuse}} \quad (\text{SOH})$$
Reciprocal: Cosecant $\csc\theta = \frac{\text{Hyp}}{\text{Opp}} = \frac{1}{\sin\theta}$.
Cosine Ratio ($\cos\theta$)
$$\cos\theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} \quad (\text{CAH})$$
Reciprocal: Secant $\sec\theta = \frac{\text{Hyp}}{\text{Adj}} = \frac{1}{\cos\theta}$.
Tangent Ratio ($\tan\theta$)
$$\tan\theta = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{\sin\theta}{\cos\theta} \quad (\text{TOA})$$
Reciprocal: Cotangent $\cot\theta = \frac{\text{Adj}}{\text{Opp}} = \frac{1}{\tan\theta}$.
§ 10.5
Special Right Triangles ($45^\circ$-$45^\circ$-$90^\circ$ & $30^\circ$-$60^\circ$-$90^\circ$)
| Special Triangle | Side Length Ratio | $\sin\theta$ Exact | $\cos\theta$ Exact | $\tan\theta$ Exact |
|---|---|---|---|---|
| $45^\circ$-$45^\circ$-$90^\circ$ (Isosceles Right) | $x : x : x\sqrt{2}$ | $\sin(45^\circ) = \frac{\sqrt{2}}{2}$ | $\cos(45^\circ) = \frac{\sqrt{2}}{2}$ | $\tan(45^\circ) = 1$ |
| $30^\circ$-$60^\circ$-$90^\circ$ ($30^\circ$ Angle) | $x : x\sqrt{3} : 2x$ | $\sin(30^\circ) = \frac{1}{2}$ | $\cos(30^\circ) = \frac{\sqrt{3}}{2}$ | $\tan(30^\circ) = \frac{\sqrt{3}}{3}$ |
| $30^\circ$-$60^\circ$-$90^\circ$ ($60^\circ$ Angle) | $x : x\sqrt{3} : 2x$ | $\sin(60^\circ) = \frac{\sqrt{3}}{2}$ | $\cos(60^\circ) = \frac{1}{2}$ | $\tan(60^\circ) = \sqrt{3}$ |
§ 10.6
Circle Geometry: Chords, Tangents & Inscribed Angles
Inscribed Angle Theorem
$$\angle_{\text{inscribed}} = \frac{1}{2} \cdot \text{Arc Measure}$$
An angle inscribed in a semicircle is strictly a right angle ($90^\circ$).
Chord Segment Products
$$a \cdot b = c \cdot d$$
Products of intersecting chord segments inside a circle are equal.
Radius-Tangent Perpendicularity
$$\text{Radius } \perp \text{ Tangent Line } (90^\circ)$$
Two tangents drawn from the same external point are congruent.
§ 10.7
Arc Length, Sector Area & Coordinate Circle Equation
Arc Length ($s$)
$$s = \frac{\theta}{360^\circ}(2\pi r) = r\theta \quad (\theta \text{ rad})$$
Curved length along the circumference subtended by central angle $\theta$.
Sector Area ($A_{\text{sector}}$)
$$A = \frac{\theta}{360^\circ}(\pi r^2) = \frac{1}{2}r^2\theta$$
Fractional area of circle enclosed by two radii and the arc.
Standard Equation of a Circle
$$(x - h)^2 + (y - k)^2 = r^2$$
Circle centered at coordinate $(h, k)$ with radius $r$.
§ 10.8
Coordinate Geometry Proofs & Quadrilateral Verification
Perpendicular Lines
$$m_1 \cdot m_2 = -1 \iff m_2 = -\frac{1}{m_1}$$
Slopes are opposite reciprocals. (Parallel: $m_1 = m_2$).
Segment Bisection
$$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$
Used to prove diagonals of a parallelogram bisect each other!
§ 10.9
3D Solids, Cavalieri's Principle & Density
PRINCIPLE 10.9.1
Topic: Cavalieri's Cross-Sectional Invariant
If two solids have the same height $h$ and the same cross-sectional area at every level parallel to their bases, then they have equal volumes!
$$\text{Pyramid / Cone Volume: } V = \frac{1}{3} B h \quad \vert \quad \text{Sphere Volume: } V = \frac{4}{3}\pi r^3$$
$$\text{Density: } \text{Density} = \frac{\text{Mass}}{\text{Volume}} \quad \vert \quad \text{Population Density} = \frac{\text{Population}}{\text{Area}}$$
§ 10.10
Scholia & Computational Geometry Speed Hacks
HACK I
Trigonometric Left-Hand Trick
$$\sin\theta = \frac{\sqrt{\text{Fingers Below}}}{2} \quad \vert \quad \cos\theta = \frac{\sqrt{\text{Fingers Above}}}{2}$$
Assign fingers to $0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ$. Fold finger to read exact radical fractions instantly!
HACK II
30-60-90 Short-Leg Anchor
$$\text{Short Leg } x \implies \text{Hypotenuse } = 2x, \ \text{Long Leg } = x\sqrt{3}$$
Always isolate the short leg across from $30^\circ$ first! Everything scales from that single number.
HACK III
Thales' Semicircle Right Angle Shortcut
$$\text{Inscribed angle intercepting diameter } \implies 90^\circ \text{ ALWAYS}$$
Any triangle with one side as the circle diameter is automatically a right triangle.
HACK IV
Area Scale Factor Squared Rule
$$\text{Linear Scale } k \implies \text{Area Scale } = k^2, \ \text{Volume Scale } = k^3$$
If dimensions double ($k=2$), area quadruples ($\times 4$) and volume octuples ($\times 8$)!
§ 10.11
The Grand Synoptic Tables & Complete Grade 10 / Geometry Student Cheat Sheet
Grade 10 / Geometry Master Reference Concordance
Authorized curriculum reference concordance • High School Geometry Complete
Congruence & Similarity
- Congruent ($\cong$): SSS, SAS, ASA, AAS, HL
- Similar ($\sim$): AA~, SAS~, SSS~
- CPCTC: Corresponding parts congruent
- Altitude: $h = \sqrt{pq}$
- Leg: $a = \sqrt{pc}, b = \sqrt{qc}$
Trigonometry (SOH-CAH-TOA)
- $\sin\theta = \frac{\text{Opp}}{\text{Hyp}}$ • $\csc\theta = \frac{\text{Hyp}}{\text{Opp}}$
- $\cos\theta = \frac{\text{Adj}}{\text{Hyp}}$ • $\sec\theta = \frac{\text{Hyp}}{\text{Adj}}$
- $\tan\theta = \frac{\text{Opp}}{\text{Adj}}$ • $\cot\theta = \frac{\text{Adj}}{\text{Opp}}$
- Pythagorean ID: $\sin^2\theta + \cos^2\theta = 1$
- $\tan\theta = \sin\theta / \cos\theta$
Special Triangles
- 45-45-90: $x : x : x\sqrt{2}$
- $\sin(45^\circ) = \cos(45^\circ) = \frac{\sqrt{2}}{2}$
- 30-60-90: $x : x\sqrt{3} : 2x$
- $\sin(30^\circ) = \frac{1}{2}, \cos(30^\circ) = \frac{\sqrt{3}}{2}$
- $\sin(60^\circ) = \frac{\sqrt{3}}{2}, \cos(60^\circ) = \frac{1}{2}$
Circle Geometry
- Inscribed Angle: $\frac{1}{2} \times \text{Arc}$
- Chords: $a \cdot b = c \cdot d$
- Radius $\perp$ Tangent ($90^\circ$)
- Circle Eq: $(x-h)^2 + (y-k)^2 = r^2$
- Thales: Semicircle angle $= 90^\circ$
Arcs & Sectors
- Arc Length: $s = r\theta = \frac{\theta}{360}(2\pi r)$
- Sector Area: $A = \frac{1}{2}r^2\theta = \frac{\theta}{360}(\pi r^2)$
- Radians to Deg: $\times \frac{180^\circ}{\pi}$
- Deg to Radians: $\times \frac{\pi}{180^\circ}$
- $2\pi \text{ rad} = 360^\circ$
3D Solids & Slopes
- Perpendicular: $m_1 \cdot m_2 = -1$
- Midpoint: $\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$
- Pyramid/Cone: $V = \frac{1}{3}Bh$
- Sphere: $V = \frac{4}{3}\pi r^3, SA = 4\pi r^2$
- Area Scale: $k^2$ • Volume: $k^3$