4.1 Plane Geometry: Polygons, Circles & Area Theorems

4.1 Plane Geometry: Polygons, Circles & Area Theorems

Euclidean plane geometry is a core component of the Bocconi Admission Test mathematics syllabus, accounting for 3 to 4 questions per exam session. Because the test prohibits calculators and enforces a strict pace of 90 seconds per question, candidates must avoid lengthy coordinate setups or ad-hoc trigonometric derivations where pure Euclidean theorems provide immediate solutions. Mastery of special triangle side ratios, similarity scaling laws, quadrilateral diagonal relations, and circle angle theorems allows complex geometric questions to be solved cleanly within seconds.

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1. Triangles: Classifications, Angle Theorems & Inequalities

A triangle is the fundamental building block of two-dimensional Euclidean geometry. Every convex polygon with $n$ sides can be partitioned into $n - 2$ non-overlapping triangles.

Fundamental Angle Properties

• Interior Angle Sum: The sum of interior angles in any Euclidean triangle equals $180^\circ$ ($ \pi$ radians): $\alpha + \beta + \gamma = 180^\circ$
• Exterior Angle Theorem: The measure of any exterior angle equals the sum of the measures of its two non-adjacent (remote) interior angles: $\theta_{\text{ext}} = \alpha + \beta$
• Polygon Generalization: For an $n$-sided convex polygon, the sum of interior angles is $S_n = (n - 2) \times 180^\circ$. The sum of exterior angles for any convex polygon is always $360^\circ$.

The Triangle Inequality Theorem

For any three line segments of lengths $a, b, c$ to form a non-degenerate triangle, each side must be strictly less than the sum of the other two and strictly greater than their absolute difference:

$|a - b| < c < a + b$

Bocconi Exam Trap: If a problem specifies two sides of length $5$ and $11$, the third side $x$ must satisfy $11 - 5 < x < 11 + 5 \implies 6 < x < 16$. If $x$ must be an integer, there are $15 - 7 + 1 = 9$ possible integer lengths.

Triangle Taxonomy

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2. Right Triangles and Special Angle Ratios

Right triangles form the backbone of quantitative geometry on the Bocconi test. Identifying special angle ratios eliminates the need for trigonometric calculations.

The Pythagorean Theorem and Common Triples

For legs $a, b$ and hypotenuse $c$, $a^2 + b^2 = c^2$. Memorizing primitive Pythagorean triples speeds up side evaluation:

• Primitive Triples: $(3, 4, 5)$, $(5, 12, 13)$, $(7, 24, 25)$, $(8, 15, 17)$, $(9, 40, 41)$, $(20, 21, 29)$.
• Scaled Triples: Multiples such as $(6, 8, 10)$, $(9, 12, 15)$, $(10, 24, 26)$, and $(15, 20, 25)$ appear frequently.

Special Right Triangles

1. $45^\circ$-$45^\circ$-$90^\circ$ Triangle (Isosceles Right Triangle):

   • Obtained by bisecting a square along its diagonal.
   • Side ratio: $1 : 1 : \sqrt{2}$.
   • Given leg $x$: hypotenuse is $x\sqrt{2}$. Given hypotenuse $d$: leg is $\frac{d}{\sqrt{2}} = \frac{d\sqrt{2}}{2}$.

2. $30^\circ$-$60^\circ$-$90^\circ$ Triangle:

   • Obtained by bisecting an equilateral triangle with an altitude.
   • Side ratio: $1 : \sqrt{3} : 2$.
   • The side opposite $30^\circ$ is $x$ (the short leg).
   • The hypotenuse opposite $90^\circ$ is $2x$ (always twice the short leg).
   • The side opposite $60^\circ$ is $x\sqrt{3}$ (the long leg).

Altitude to the Hypotenuse & Euclid's Theorems

Let $\triangle ABC$ have a right angle at $C$, with altitude $h = CH$ dropped to hypotenuse $c = AB$. The altitude divides the hypotenuse into two segments $p = AH$ and $q = HB$ such that $c = p + q$:

1. First Theorem of Euclid: Each leg is the geometric mean of the hypotenuse and its adjacent projection onto the hypotenuse: $a^2 = c \cdot q \qquad \text{and} \qquad b^2 = c \cdot p$
2. Second Theorem of Euclid: The altitude is the geometric mean of the two segments of the hypotenuse: $h^2 = p \cdot q \iff h = \sqrt{p \cdot q}$
3. Area Invariance for Altitude: Equating two expressions for triangle area gives: $\text{Area} = \frac{1}{2} a b = \frac{1}{2} c h \implies h = \frac{a b}{c}$

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3. Congruence, Similarity & Area Scaling Laws

Congruence Criteria

Two triangles are congruent (identical in shape and size, $\triangle ABC \cong \triangle A'B'C'$) if they satisfy any of the following:

• SSS (Side-Side-Side): All three pairs of corresponding sides are equal.
• SAS (Side-Angle-Side): Two sides and the included angle are equal.
• ASA (Angle-Side-Angle) / AAS: Two angles and any corresponding side are equal.
• HL (Hypotenuse-Leg): In right triangles, hypotenuse and one leg are equal.

Similarity Criteria

Two triangles are similar (identical in shape, $\triangle ABC \sim \triangle A'B'C'$) if corresponding angles are equal and corresponding sides are proportional:

• AA (Angle-Angle): Two corresponding angles are equal.
• SAS Similarity: One pair of equal angles flanked by proportional sides.
• SSS Similarity: All three pairs of corresponding sides are in the same ratio $k$.

The Fundamental Dimensional Scaling Laws

When two figures are geometrically similar with linear scale factor $k = \frac{\ell_2}{\ell_1}$:

• Perimeter Ratio: $\frac{P_2}{P_1} = k$
• Area Ratio: $\frac{A_2}{A_1} = k^2$
• Volume Ratio (3D Solids): $\frac{V_2}{V_1} = k^3$

Bocconi Exam Trap: If the linear dimensions of a polygon increase by $40%$, the scale factor is $k = 1.4$. The area increases by $k^2 = (1.4)^2 = 1.96$, which is an increase of $96%$, not $40%$ or $80%$.

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4. Comprehensive Triangle Area Formulas

Depending on the given parameters, compute triangle area using the most direct formula:

1. Standard Base-Height: $A = \frac{1}{2} b h$
2. Trigonometric (Two Sides and Included Angle): $A = \frac{1}{2} a b \sin(C) = \frac{1}{2} b c \sin(A) = \frac{1}{2} a c \sin(B)$
3. Heron's Formula (Three Known Sides): For side lengths $a, b, c$, first compute the semi-perimeter $s$: $s = \frac{a + b + c}{2}$ $A = \sqrt{s(s - a)(s - b)(s - c)}$
4. Inradius Formula ($r$): $A = r \cdot s \implies r = \frac{A}{s}$
5. Circumradius Formula ($R$): $A = \frac{a b c}{4R} \implies R = \frac{a b c}{4A}$

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5. Quadrilaterals: Taxonomy, Symmetry & Formulas

Quadrilaterals are 4-sided closed polygons with interior angle sum $(4 - 2) \times 180^\circ = 360^\circ$.

Properties of Standard Quadrilaterals

• Parallelogram:

  • Opposite sides are parallel and equal: $AB \parallel CD$, $AD \parallel BC$, $AB = CD$, $AD = BC$.
  • Opposite angles are equal; consecutive angles are supplementary ($\alpha + \beta = 180^\circ$).
  • Diagonals bisect each other (their intersection is the midpoint of both).
  • Area: $A = b \cdot h = a b \sin(\theta)$.

• Rectangle:

  • Equiangular parallelogram (all four interior angles are $90^\circ$).
  • Diagonals are congruent: $d = \sqrt{w^2 + h^2}$.
  • Area: $A = w \cdot h$.

• Rhombus:

  • Equilateral parallelogram (all four sides are equal: $a = b = c = d$).
  • Diagonals are perpendicular bisectors of each other ($d_1 \perp d_2$) and bisect the interior angles.
  • Area: $A = \frac{d_1 \cdot d_2}{2} = a^2 \sin(\theta)$.
  • Side-Diagonal Identity: The four right triangles formed by the diagonals have legs $\frac{d_1}{2}$ and $\frac{d_2}{2}$ and hypotenuse $a$: $\left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2 = a^2 \iff d_1^2 + d_2^2 = 4a^2$

• Square:

  • Both a rectangle and a rhombus (regular quadrilateral).
  • Diagonals are equal, perpendicular, and bisect each other: $d = s\sqrt{2}$.
  • Area: $A = s^2 = \frac{d^2}{2}$.

• Trapezoid (Trapezium):

  • Exactly one pair of parallel sides (bases $b_1$ and $b_2$).
  • Mid-segment (median) connects the midpoints of non-parallel legs: $m = \frac{b_1 + b_2}{2}$.
  • Area: $A = \frac{b_1 + b_2}{2} \cdot h = m \cdot h$.
  • Isosceles Trapezoid: Non-parallel legs are equal; base angles are equal; diagonals are congruent.

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6. Circle Geometry: Chords, Tangents & Angle Theorems

A circle is the locus of all points in a plane equidistant from a fixed center $O$.

• Circumference: $C = 2\pi r = \pi d$.
• Area: $A = \pi r^2$.

Chords and Perpendicular Bisectors

• A chord is a line segment connecting two points on the circle. The longest chord is the diameter ($d = 2r$).
• The perpendicular dropped from the center $O$ to any chord $AB$ bisects the chord at $M$.
• In the right triangle formed by the radius $r$, the distance $d$ from the center to the chord, and the half-chord length $\frac{L}{2}$: $r^2 = d^2 + \left(\frac{L}{2}\right)^2$

Tangent Lines to a Circle

• A tangent line intersects the circle at exactly one point (the point of tangency $T$).
• Fundamental Tangent Theorem: The radius drawn to the point of tangency is strictly perpendicular to the tangent line: $OT \perp t$.
• Tangents from an External Point: Two tangents drawn to a circle from an external point $P$ have equal lengths ($PT_1 = PT_2$). The line connecting $P$ to the center $O$ bisects the angle between the tangents.

Central vs. Inscribed Angles

• Central Angle: An angle whose vertex is the circle center $O$, subtending an arc $\widehat{AB}$. Its angular measure equals the measure of the arc: $\angle AOB = \theta$.
• Inscribed Angle: An angle whose vertex lies on the circle circumference, subtending the same arc $\widehat{AB}$.
• Inscribed Angle Theorem: The measure of an inscribed angle is exactly half the measure of the central angle subtending the same arc: $\theta_{\text{inscribed}} = \frac{1}{2} \theta_{\text{central}}$
• Consequences:
  1. All inscribed angles that subtend the same arc are congruent.
  2. Thales' Theorem: Any angle inscribed in a semicircle subtends a diameter and is a right angle ($90^\circ$).

Arc Length and Sector Area

For a central angle $\theta$ subtending an arc in a circle of radius $r$:

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7. Worked Step-by-Step Examples

Example 1: Heron's Formula and Inradius

Calculate the area and inradius of a triangle with side lengths $a = 13, b = 14, c = 15$.

1. Compute semi-perimeter: $s = \frac{13 + 14 + 15}{2} = \frac{42}{2} = 21$.
2. Compute differences: $s - a = 21 - 13 = 8$, $s - b = 21 - 14 = 7$, $s - c = 21 - 15 = 6$.
3. Apply Heron's formula: $A = \sqrt{21 \cdot 8 \cdot 7 \cdot 6} = \sqrt{(3 \cdot 7) \cdot (8) \cdot (7) \cdot (2 \cdot 3)} = \sqrt{7^2 \cdot 3^2 \cdot 16} = 7 \cdot 3 \cdot 4 = 84$
4. Compute inradius: $r = \frac{A}{s} = \frac{84}{21} = 4$.

Example 2: Rhombus Diagonal and Area Relations

A rhombus has a side length of $10$ and its area is $96$. Determine the lengths of its two diagonals.

1. Area formula: $\frac{d_1 d_2}{2} = 96 \implies d_1 d_2 = 192$.
2. Side-diagonal relation: $d_1^2 + d_2^2 = 4a^2 = 4(10^2) = 400$.
3. Construct algebraic identities:
   • $(d_1 + d_2)^2 = d_1^2 + d_2^2 + 2d_1 d_2 = 400 + 2(192) = 400 + 384 = 784 \implies d_1 + d_2 = \sqrt{784} = 28$.
   • $(d_1 - d_2)^2 = d_1^2 + d_2^2 - 2d_1 d_2 = 400 - 384 = 16 \implies d_1 - d_2 = \sqrt{16} = 4$.
4. Add equations: $2d_1 = 32 \implies d_1 = 16$, and $d_2 = 28 - 16 = 12$. The diagonals are $16$ and $12$.

Example 3: Inscribed vs. Central Angles and Chords

A chord $AB$ of length $6\sqrt{2}$ lies in a circle of radius $6$. Find the measure of the inscribed angle subtending minor arc $\widehat{AB}$.

1. In $\triangle OAB$, $OA = OB = 6$, and $AB = 6\sqrt{2}$.
2. Notice $OA^2 + OB^2 = 6^2 + 6^2 = 36 + 36 = 72 = (6\sqrt{2})^2 = AB^2$.
3. By the converse of the Pythagorean theorem, $\triangle OAB$ is a right isosceles triangle with central angle $\angle AOB = 90^\circ$.
4. By the Inscribed Angle Theorem, the inscribed angle subtending $\widehat{AB}$ is $\frac{1}{2} \angle AOB = \frac{90^\circ}{2} = 45^\circ$.