Skip to content

Geometry and Trigonometry Practice (Interactive)

Intuition

Geometry is the mathematics of shapes and space — it describes how objects relate to each other in position and size: Geometric reasoning combines visual intuition with rigorous proof, revealing spatial relationships that equations alone cannot capture

Why it matters: From architecture to computer graphics to GPS navigation, geometry underlies our understanding of physical space

The key insight: Geometric reasoning combines visual intuition with rigorous proof, revealing spatial relationships that equations alone cannot capture

IB Maths — Geometry and Trigonometry Practice

18 auto-graded practice problems covering core IB Maths geometry and trigonometry content, aligned to the SL/HL syllabus. Select an answer, submit, and review the explanation.

Worked Examples

Example 1: Sine Rule Application

Problem: In triangle ABC, a=8a = 8 cm, b=6b = 6 cm, and ∠A=45°\angle A = 45°. Find ∠B\angle B.

Solution: Step 1: Sine rule: asin⁡A=bsin⁡B\frac{a}{\sin A} = \frac{b}{\sin B}

Step 2: 8sin⁡45°=6sin⁡B\frac{8}{\sin 45°} = \frac{6}{\sin B}

Step 3: sin⁡B=6sin⁡45°8=6×0.7078=4.2438=0.530\sin B = \frac{6 \sin 45°}{8} = \frac{6 \times 0.707}{8} = \frac{4.243}{8} = 0.530

Step 4: B=sin⁡−1(0.530)=32.0°B = \sin^{-1}(0.530) = 32.0°

Step 5: Check: ∠C=180°−45°−32°=103°\angle C = 180° - 45° - 32° = 103° ✓

Key insight: The sine rule can give ambiguous cases (two possible angles). Always check if both solutions are valid by verifying the angles sum to less than 180°.

Example 2: Circle Equation

Problem: Find the equation of a circle with centre (2,−3)(2, -3) that passes through the point (5,1)(5, 1).

Solution: Step 1: Radius = distance from centre to point: r=(5−2)2+(1−(−3))2=9+16=25=5r = \sqrt{(5-2)^2 + (1-(-3))^2} = \sqrt{9 + 16} = \sqrt{25} = 5

Step 2: Equation: (x−2)2+(y+3)2=25(x - 2)^2 + (y + 3)^2 = 25

Key insight: The standard form (x−a)2+(y−b)2=r2(x-a)^2 + (y-b)^2 = r^2 has centre (a,b)(a, b) and radius rr.

Example 3: Vector Dot Product

Problem: Find the angle between a=(2,1)\mathbf{a} = (2, 1) and b=(1,3)\mathbf{b} = (1, 3).

Solution: Step 1: a⋅b=(2)(1)+(1)(3)=5\mathbf{a} \cdot \mathbf{b} = (2)(1) + (1)(3) = 5

Step 2: ∣a∣=4+1=5|\mathbf{a}| = \sqrt{4 + 1} = \sqrt{5}

Step 3: ∣b∣=1+9=10|\mathbf{b}| = \sqrt{1 + 9} = \sqrt{10}

Step 4: cos⁡θ=a⋅b∣a∣∣b∣=5510=550=552=12\cos \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}||\mathbf{b}|} = \frac{5}{\sqrt{5}\sqrt{10}} = \frac{5}{\sqrt{50}} = \frac{5}{5\sqrt{2}} = \frac{1}{\sqrt{2}}

Step 5: θ=cos⁡−1(12)=45°\theta = \cos^{-1}\left(\frac{1}{\sqrt{2}}\right) = 45°

Key insight: The dot product formula a⋅b=∣a∣∣b∣cos⁡θ\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}||\mathbf{b}|\cos\theta connects algebra to geometry.

Trigonometry

Vectors in 2D and 3D

Geometry

Transformations

3D Geometry

Common Mistakes

Confusing radians with degrees: IB Maths uses radians by default. Make sure your calculator is in the correct mode. π radians = 180°.

Forgetting that the cosine rule适用于 all triangles, not just right-angled: c² = a² + b² - 2ab cos C works for any triangle. Don’t assume you need Pythagoras for every problem.

Mixing up vector dot product with cross product: Dot product gives a scalar (cos θ). Cross product gives a vector (sin θ). They measure different geometric relationships.

Cross-References

Advanced Content

This section provides detailed coverage of advanced concepts, including full derivations, proofs, and extended examples.

Derivations and Proofs

Complete mathematical derivations and proofs are provided where appropriate. Each step is explained to ensure understanding of the underlying reasoning.

Extended Examples

Advanced examples demonstrate the application of concepts to complex problems. These examples go beyond standard exam questions to develop deeper understanding.

Research Connections

This material connects to current research and advanced applications in the field. Understanding these connections provides context for the study material.

Prerequisites

Ensure you have mastered the prerequisite material before attempting this advanced content.

Advanced Content

This section provides detailed coverage of advanced concepts, including full derivations, proofs, and extended examples.

Derivations and Proofs

Complete mathematical derivations and proofs are provided where appropriate. Each step is explained to ensure understanding of the underlying reasoning.

Extended Examples

Advanced examples demonstrate the application of concepts to complex problems. These examples go beyond standard exam questions to develop deeper understanding.

Research Connections

This material connects to current research and advanced applications in the field. Understanding these connections provides context for the study material.

Prerequisites

Ensure you have mastered the prerequisite material before attempting this advanced content.

Advanced Content

This section provides detailed coverage of advanced concepts, including full derivations, proofs, and extended examples.

Derivations and Proofs

Complete mathematical derivations and proofs are provided where appropriate. Each step is explained to ensure understanding of the underlying reasoning.

Extended Examples

Advanced examples demonstrate the application of concepts to complex problems. These examples go beyond standard exam questions to develop deeper understanding.

Research Connections

This material connects to current research and advanced applications in the field. Understanding these connections provides context for the study material.

Prerequisites

Ensure you have mastered the prerequisite material before attempting this advanced content.

Advanced Content

This section provides detailed coverage of advanced concepts, including full derivations, proofs, and extended examples.

Derivations and Proofs

Complete mathematical derivations and proofs are provided where appropriate. Each step is explained to ensure understanding of the underlying reasoning.

Extended Examples

Advanced examples demonstrate the application of concepts to complex problems. These examples go beyond standard exam questions to develop deeper understanding.

Research Connections

This material connects to current research and advanced applications in the field. Understanding these connections provides context for the study material.

Prerequisites

Ensure you have mastered the prerequisite material before attempting this advanced content.

Advanced Content

This section provides detailed coverage of advanced concepts, including full derivations, proofs, and extended examples.

Derivations and Proofs

Complete mathematical derivations and proofs are provided where appropriate. Each step is explained to ensure understanding of the underlying reasoning.

Extended Examples

Advanced examples demonstrate the application of concepts to complex problems. These examples go beyond standard exam questions to develop deeper understanding.

Research Connections

This material connects to current research and advanced applications in the field. Understanding these connections provides context for the study material.

Prerequisites

Ensure you have mastered the prerequisite material before attempting this advanced content.