4MA1/1HR Jan 2019, Question 20: Trigonometry in 3D: Cuboids
Pearson Edexcel International GCSE Mathematics A 4MA1 · Paper 1HR (Higher) · Calculator · 4 marks
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The question, in short
Cube ABCDEFGH with M the midpoint of GH; find the angle between line MA and the plane ABCD, giving the answer to 1 decimal place.
Summarised in our own words. Use the official paper for the exact wording and the diagram.
Solution · 4 marks
- 1Let the edge of the cube be 2a
- 2Let N be the midpoint of BC, the point of plane ABCD directly below M
- 3BN = a and AB = 2a, and AB is perpendicular to BC
- 4AN² = AB² + BN² = (2a)² + a² = 4a² + a² = 5a², so AN = √5 a
- 5MN is vertical and equal to an edge of the cube, so MN = 2a
- 6The angle between MA and the plane ABCD is angle MAN
- 7tan(MAN) = MN ÷ AN = 2a ÷ √5 a = 2/√5 = 0.8944...
- 8MAN = tan⁻¹(0.8944...) = 41.81...°
- 9Angle = 41.8° (1 d.p.)
Answer41.8
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