https://file.aiquickdraw.com/mathbot/file/1725965404278hyl5do2u.jpg

Let's solve the two problems shown in the image.

### Problem 10: Finding the Angle Between the Diagonals of a Rectangle

**Given:**
- Sides of the rectangle are 3 cm and 4 cm.

**Solution:**
The diagonals of a rectangle intersect at right angles, meaning the angle between the diagonals can be found using trigonometric identities.

For a rectangle with sides \( a = 3 \, \text{cm} \) and \( b = 4 \, \text{cm} \), the length of the diagonal \( d \) can be calculated using the Pythagorean theorem:
\[
d = \sqrt{a^2 + b^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \, \text{cm}
\]

Since the diagonals bisect each other and form right triangles, the angle \( \theta \) between the diagonals can be found using the tangent function:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{a}{b} = \frac{3}{4}
\]

Thus, the angle \( \theta \) is:
\[
\theta = \tan^{-1}\left(\frac{3}{4}\right)
\]
Calculating this gives:
\[
\theta \approx 36.87^\circ
\]
So, the angle between the diagonals is approximately \( 36.87^\circ \).

### Problem 11: Finding the Length of the Flagstaff

**Given:**
- Distance from the tower = 60 m
- Angle of elevation to the top of the flagstaff = 64°
- Angle of elevation to the bottom of the flagstaff = 62°

**Solution:**

Let's denote:
- \( h_1 \) as the height of the tower (up to the bottom of the flagstaff).
- \( h_2 \) as the height of the flagstaff alone.
- \( h_1 + h_2 \) as the total height from the ground to the top of the flagstaff.

Using the tangent function for the angles of elevation:
\[
\tan(64^\circ) = \frac{h_1 + h_2}{60}
\]
\[
\tan(62^\circ) = \frac{h_1}{60}
\]

This gives us two equations:
\[
h_1 + h_2 = 60 \cdot \tan(64^\circ)
\]
\[
h_1 = 60 \cdot \tan(62^\circ)
\]

Subtract the second equation from the first to find \( h_2 \):
\[
h_2 = 60 \cdot (\tan(64^\circ) - \tan(62^\circ))
\]

Now, calculating the values:
\[
h_2 \approx 60 \cdot (2.0503 - 1.8807) = 60 \cdot 0.1696 = 10.176 \, \text{m}
\]

So, the length of the flagstaff is approximately **10.18 meters**.

---

Do you want any further details or explanations? Here are 5 related questions:

1. How would the angle between the diagonals change if the sides of the rectangle were different?
2. Can you derive the relationship between the sides and diagonals of a rectangle geometrically?
3. What would be the length of the flagstaff if the angles were closer together?
4. How would the solution change if the observer was further from the tower?
5. Can you apply these trigonometric principles to find the height of a building using a similar method?

**Tip:** Understanding the tangent function and its applications in trigonometry can simplify many real-world height and distance problems.

10. Find the angle between the diagonals of a rectangle whose sides are 3 cm and 4 cm. 11. A man standing 60 m away from a tower notices that the angles of elevation of the top and bottom of a flagstaff on top of the tower are 64° and 62° respectively. Find the length of the flagstaff.

Solve two geometry problems: finding the angle between the diagonals of a rectangle with sides 3 cm and 4 cm, and calculating the length of a flagstaff based on angles of elevation. This problem uses the Pythagorean Theorem and trigonometric functions such as tangent.

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