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Solve where AD = 28m, AE = 41m, DE = 22m, and BC = 64m

The problem involves a geometric diagram with triangle \(ABC\) and points \(D\) and \(E\) located on sides \(AC\) and \(AB\), respectively, with \(DE\) as a straight line. The given information is:

- \(AD = 28 \, \text{m}\)
- \(AE = 41 \, \text{m}\)
- \(DE = 22 \, \text{m}\)
- \(BC = 64 \, \text{m}\)

The tasks are:

1. Calculate the size of angle \(DAE\).
2. Calculate the length \(CD\).

### 1. Calculate the size of angle \(DAE\):

To find angle \(DAE\), we can use the cosine rule in triangle \(ADE\), where:

- \(AD = 28 \, \text{m}\)
- \(AE = 41 \, \text{m}\)
- \(DE = 22 \, \text{m}\)

The cosine rule is given by:

\[
\cos(\theta) = \frac{AD^2 + AE^2 - DE^2}{2 \times AD \times AE}
\]

Substituting the values:

\[
\cos(\theta) = \frac{28^2 + 41^2 - 22^2}{2 \times 28 \times 41}
\]

\[
\cos(\theta) = \frac{784 + 1681 - 484}{2296}
\]

\[
\cos(\theta) = \frac{1981}{2296}
\]

\[
\cos(\theta) \approx 0.8627
\]

\[
\theta \approx \cos^{-1}(0.8627)
\]

\[
\theta \approx 30.47^\circ
\]

So, angle \(DAE\) is approximately \(30.47^\circ\).

### 2. Calculate the length \(CD\):

From the diagram, \(D\) and \(E\) form a straight line with \(AC\). Hence, using the triangle proportionality theorem (since \(DE\) is parallel to \(BC\)), we can set up the proportion:

\[
\frac{AD}{AC} = \frac{DE}{BC}
\]

Given:

- \(AD = 28 \, \text{m}\)
- \(DE = 22 \, \text{m}\)
- \(BC = 64 \, \text{m}\)

Let's find \(AC\):

\[
\frac{28}{AC} = \frac{22}{64}
\]

\[
28 \times 64 = 22 \times AC
\]

\[
AC = \frac{28 \times 64}{22}
\]

\[
AC \approx 81.09 \, \text{m}
\]

To find \(CD\), note:

\[
CD = AC - AD
\]

\[
CD = 81.09 - 28
\]

\[
CD \approx 53.09 \, \text{m}
\]

Therefore, the length of \(CD\) is approximately \(53.09 \, \text{m}\).

### Summary

1. The size of angle \(DAE\) is approximately \(30.47^\circ\).
2. The length \(CD\) is approximately \(53.09 \, \text{m}\).

Would you like further details or have any questions?

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Here are 5 related questions to deepen understanding:

1. What is the significance of the cosine rule in finding angles in non-right-angled triangles?
2. How does the triangle proportionality theorem apply to triangles with parallel lines?
3. Could this problem be solved using trigonometric ratios like sine or tangent instead of cosine?
4. What other methods could be used to calculate the length of \(CD\) in triangle \(ADE\)?
5. How would the solution change if \(DE\) were not parallel to \(BC\)?

**Tip:** When using the cosine rule, ensure all sides of the triangle are known or can be calculated to accurately find the angle or side length.

Learn how to solve a geometric triangle problem involving sides AD = 28m, AE = 41m, DE = 22m, and BC = 64m. Calculate angle DAE and length CD using the cosine rule and triangle proportionality theorem. Suitable for high school students.

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