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The diagram shows a triangular prism - Edexcel - GCSE Maths - Question 19 - 2019 - Paper 2
Question 19
The diagram shows a triangular prism. The base, ABCD, of the prism is a square of side length 15 cm. Angle ABE and angle CBE are right angles. Angle EAB = 35°. M i... show full transcript
Worked Solution & Example Answer:The diagram shows a triangular prism - Edexcel - GCSE Maths - Question 19 - 2019 - Paper 2
Step 1
Calculate DM and MA
Answer
Given the ratio DM:MA = 2:3, let DM = 2x and MA = 3x.
Since DA = 15 cm (the length of the square), we can set up the equation:
[ DM + MA = DA \implies 2x + 3x = 15 \implies 5x = 15 \implies x = 3 ]\n Thus, we have: [ DM = 2x = 6 \text{ cm} ]; [ MA = 3x = 9 \text{ cm} ]
Step 2
Step 3
Step 4
Calculate the angle between EM and the base
Answer
To find the angle θ between EM and the base (ABCD), we will use the dot product between vector EM and vector normal to the base.
Vector EM = E - M = (12.29, 8.59, 15) - (0, 0, 6) = (12.29, 8.59, 9).
The normal vector to the base ABCD can be taken as (0, 0, 1). Thus:
Using the dot product: [ EM . (0, 0, 1) = 9 ]
Magnitude of EM: [ |EM| = \sqrt{(12.29)^2 + (8.59)^2 + (9)^2} \approx 15.71 ]
Now, use cosine: [ \cos(θ) = \frac{9}{15.71} \implies θ = \cos^{-1}\left(\frac{9}{15.71}\right) \approx 40.7° ]
Thus, the angle between EM and the base of the prism is approximately (40.7°).