The diagram shows a triangular prism. The base, ABCD, of the prism is a square of side length 15 cm. Angle ABC and angle CBE are right angles. Angle EAB = 35° M is the point on DA such that DM:MA = 2:3 Calculate the size of the angle between EM and the base of the prism. Give your answer correct to 1 decimal place.

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The diagram shows a triangular prism - Edexcel - GCSE Maths - Question 1 - 2019 - Paper 3

Question 1

The diagram shows a triangular prism. The base, ABCD, of the prism is a square of side length 15 cm. Angle ABC 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 1 - 2019 - Paper 3

Step 1

Calculate the length of DM and MA

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Answer

Since DM:MA = 2:3, we can denote the lengths as follows:

Let the total length of DA be divided into 5 equal parts: 2 parts for DM and 3 parts for MA.
The length of DA is 15 cm. Thus, each part is
$\frac{15 \text{ cm}}{5} = 3 \text{ cm}$ .
Therefore, DM = 2 parts = 2 * 3 cm = 6 cm, and MA = 3 parts = 3 * 3 cm = 9 cm.

Step 2

Determine coordinates of points

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Assuming A is at (0, 0, 0), B at (15, 0, 0), C at (15, 15, 0), and D at (0, 15, 0):

Point M is located on DA, specifically at (0, 6, 0) since DM = 6 cm.
Point E, considering the angle EAB of 35°, can be derived.
Using trigonometry, coordinates of E can be calculated based on height (h):
$x = 15 + 15 * \cos (35°)$
$y = 15 * \sin (35°)$ .

Step 3

Calculate the angle between EM and the base

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Answer

To find the angle EM makes with the base (the plane formed by ABCD), we use the tangent function to establish:

Let the height of E above the base, which is the vertical distance (h), be calculated:
$h = 15 * \sin(35°).$
From the coordinates of M and E, calculate the slope of line EM. The angle θ can be found from:
Using the inverse tangent function: $\tan(θ) = \frac{h}{\text{horizontal distance between M and E}} = \tan(θ) = \frac{h}{15 + 15 * \cos(35°) - 0}.$
Finding θ gives the required angle with respect to the base.

The diagram shows a triangular prism - Edexcel - GCSE Maths - Question 1 - 2019 - Paper 3

Worked Solution & Example Answer:The diagram shows a triangular prism - Edexcel - GCSE Maths - Question 1 - 2019 - Paper 3

Calculate the length of DM and MA

Determine coordinates of points

Calculate the angle between EM and the base

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