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In the rectangular prism shown, AD = 7 cm, AE = 8 cm, EF = 6 cm - HSC - SSCE Mathematics Advanced - Question 22 - 2023 - Paper 1

Question 22

In the rectangular prism shown, AD = 7 cm, AE = 8 cm, EF = 6 cm. Point M is the midpoint of CD. Find ∠AEM, to the nearest degree.

Worked Solution & Example Answer:In the rectangular prism shown, AD = 7 cm, AE = 8 cm, EF = 6 cm - HSC - SSCE Mathematics Advanced - Question 22 - 2023 - Paper 1

Step 1

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Answer

To determine the length of AM, we can use the Pythagorean theorem in triangle ADM:

Given:

Now apply the Pythagorean theorem:

$AM^2 = AD^2 + DM^2$

Substituting the values:

$AM^2 = 7^2 + 3^2 = 49 + 9 = 58$

So,

$AM = ext{sqrt}(58)$ .

Step 2

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Answer

Now we analyze triangle AEM to find the angle ∠AEM (denote this angle as α). We have:

Using the tangent function, we get:

$an(α) = \frac{AM}{AE} = \frac{\sqrt{58}}{8}$

Calculating α:

$α = \tan^{-1}(\frac{\sqrt{58}}{8}) \approx 43.59°$

Thus,

So, rounding to the nearest degree:

$α ≈ 44°$ .

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