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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