Geometry Basics
🟢 Lite — Quick Review (1h–1d)
Rapid summary for last-minute revision before your exam.
Geometry deals with shapes, sizes, and the properties of space. In the UI entrance test, geometry questions test your understanding of angles, areas, perimeters, and the relationships between 2D shapes. Master the key angle rules and area/perimeter formulas.
Essential Angle Rules:
- Angles on a straight line sum to 180°
- Angles around a point sum to 360°
- Vertically opposite angles are equal
- In a triangle, angles sum to 180°; in a quadrilateral, to 360°
- Parallel lines: corresponding angles are equal; alternate angles are equal; co-interior (allied) angles sum to 180°
Key Area and Perimeter Formulas:
- Rectangle: area = l × w; perimeter = 2(l + w)
- Triangle: area = ½ × base × height
- Circle: area = πr²; circumference = 2πr
- Trapezium: area = ½(a + b)h, where a, b are parallel sides
- Parallelogram: area = base × height
Pythagoras’ Theorem: In a right-angled triangle, a² + b² = c², where c is the hypotenuse. Common Pythagorean triples: (3,4,5), (5,12,13), (8,15,17), (7,24,25).
⚡ Exam Tip: In a parallel-line geometry problem, look for F-shaped (corresponding), Z-shaped (alternate), or C-shaped (co-interior) angle configurations. For area problems with composite shapes, split the shape into rectangles, triangles, and circles, calculate each area separately, then add or subtract as appropriate.
🟡 Standard — Regular Study (2d–2mo)
Standard content for students with a few days to months.
Basic Angles
Angles on a straight line: If one angle is 120°, the adjacent angle on the same line is 60°. Vertically opposite angles: When two lines intersect, opposite angles are equal. If one angle is 65°, the opposite angle is also 65°.
Parallel Lines
When a transversal crosses two parallel lines:
- Corresponding angles (same position, e.g., top-left of each intersection) are equal: ∠ in top-left at first intersection = ∠ in top-left at second intersection
- Alternate angles (on opposite sides of the transversal, inside the parallel lines) are equal
- Co-interior/allied angles (same side of transversal, inside parallel lines) sum to 180°
Triangles
Angle sum: All three interior angles sum to 180°. Exterior angle: An exterior angle equals the sum of the two opposite interior angles. Isosceles triangle: Two sides equal → two angles opposite those sides are equal. Equilateral triangle: All sides equal, all angles = 60°.
Example: In triangle ABC, ∠A = 50° and ∠B = 60°. Find ∠C. ∠C = 180° − 50° − 60° = 70°.
Pythagoras’ Theorem
For right-angled triangles only: the square of the hypotenuse equals the sum of squares of the other two sides.
Example: A ladder 5 m long leans against a wall, with its foot 3 m from the wall. How high does it reach? 3² + h² = 5² → 9 + h² = 25 → h² = 16 → h = 4 m.
Quadrilaterals
Parallelogram: opposite sides parallel and equal; opposite angles equal; diagonals bisect each other. Area = base × height. Rectangle: parallelogram with all angles 90°. Diagonal d = √(l² + w²). Rhombus: all sides equal; opposite angles equal; diagonals are perpendicular bisectors. Trapezium: one pair of parallel sides. Area = ½(sum of parallel sides) × height. Kite: two pairs of adjacent sides equal; one pair of opposite angles equal; diagonals are perpendicular.
Circles
Circumference = 2πr = πd. Area = πr². Arc length = (θ/360) × 2πr. Sector area = (θ/360) × πr². Chord: a line segment connecting two points on the circle. Perpendicular bisector of a chord passes through the centre.
Similar Figures
Two shapes are similar if they have the same shape but different sizes (corresponding angles equal, corresponding sides in proportion). Scale factor = (length of side in image) / (corresponding side in original). Area scale factor = (scale factor)². Volume scale factor = (scale factor)³.
Problem-Solving Strategies:
- For complex composite shapes, draw the shape and label all known dimensions
- In right-angled triangles, check whether Pythagorean triples apply before doing long calculations
- For similar triangles, identify which sides correspond and set up a ratio equation
- In circle problems, draw radii to vertices — this often creates isosceles triangles you can work with
Common Mistakes:
- Using Pythagoras’ theorem on non-right-angled triangles — it only applies to right-angled triangles
- Confusing diameter with radius in circle calculations
- Forgetting that in an isosceles triangle, the altitude from the apex also bisects the base
- In similar figure problems, mixing up the scale factor for area (square it!) or volume (cube it!)
🔴 Extended — Deep Study (3m3+)
Comprehensive coverage for students on a longer study timeline.
Angle Chasing in Complex Diagrams
Angle chasing involves using known angle relationships to find unknown angles step by step.
Example: In triangle ABC, D is on AC. Lines AB and CD are extended to meet at E, with AB ⟂ CD. If ∠DBC = 30° and ∠BCA = 50°, find ∠CDE. ∠BCD = 50° (alternate interior, AB ∥ CD? Wait, need to check configuration. Assuming AB is a line through B intersecting CD at right angle at some point — draw the diagram.) Using the rule: exterior angle of a triangle = sum of opposite interior angles. ∠CDE is an exterior angle of triangle BCD (if D is on line extended from C). So ∠CDE = ∠DBC + ∠BCD = 30° + 50° = 80°. ✓
Proof: Sum of Interior Angles of a Polygon
An n-sided polygon can be divided into (n − 2) triangles by drawing diagonals from one vertex. Each triangle has angles summing to 180°. So sum of interior angles = (n − 2) × 180°. Example: A hexagon (n=6): (6−2) × 180° = 4 × 180° = 720°. Exterior angles always sum to 360° regardless of the number of sides.
Circumcircle and Incircle of Triangles
The circumcircle passes through all three vertices of a triangle. Its centre is the intersection of perpendicular bisectors of the sides. Radius R = abc / (4Δ), where Δ is the area of the triangle. The incircle is tangent to all three sides. Its centre is the intersection of angle bisectors. Radius r = Δ/s, where s = semi-perimeter = (a+b+c)/2.
Properties of Circle Theorems
Key theorems (assumed from standard geometry):
- Angle subtended by a diameter at the circumference = 90° (Thales’ theorem).
- Angles in the same segment are equal.
- The angle at the centre is twice the angle at the circumference subtended by the same arc.
- Opposite angles of a cyclic quadrilateral sum to 180°.
- Tangent to a circle is perpendicular to the radius at the point of contact.
- Two tangents from an external point are equal in length.
Coordinate Geometry — Basics
Distance between two points (x₁, y₁) and (x₂, y₂): d = √((x₂−x₁)² + (y₂−y₁)²). Midpoint: ((x₁+x₂)/2, (y₁+y₂)/2). Gradient of line: m = (y₂−y₁)/(x₂−x₁). Equation of line through (x₁, y₁) with gradient m: y − y₁ = m(x − x₁). Parallel lines have equal gradients. Perpendicular lines have gradients m₁ × m₂ = −1.
Locus Problems
A locus is the set of all points satisfying a condition.
- Points equidistant from two fixed points: perpendicular bisector of the segment joining them.
- Points at fixed distance r from a fixed point: circle of radius r centred at that point.
- Points equidistant from two intersecting lines: the angle bisectors (two lines at 90° to each other bisecting the angles).
UI Entrance Exam Patterns
Geometry questions in the UI Academic Potential test include:
- Angle calculation using basic rules
- Parallel line angle problems
- Triangle angle problems
- Pythagoras’ theorem applications
- Perimeter and area calculations
- Similar figures
- Basic circle geometry
⚡ Exam Strategy: When an angle seems impossible to find directly, look for an exterior angle relationship or a cyclic quadrilateral property. Adding an auxiliary line (e.g., extending a side, drawing a diagonal) often creates the triangle or parallel line configuration needed.
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