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### Problem 4: Area of the Rhombus

A rhombus has sides of length 5 cm and an angle of 65°. To find the area, we can use the formula for the area of a rhombus when given the side length \( a \) and one angle \( \theta \):
\[
\text{Area} = a^2 \cdot \sin(\theta)
\]
where:
- \( a = 5 \, \text{cm} \) (side length),
- \( \theta = 65^\circ \) (the angle between two sides),
- \( \sin(65^\circ) \approx 0.9063 \).

Now, plugging in the values:
\[
\text{Area} = 5^2 \cdot \sin(65^\circ) = 25 \cdot 0.9063 \approx 22.66 \, \text{cm}^2
\]

So, the area of the rhombus is approximately **22.66 cm²**.

---

### Problem 5: Finding the remaining side of the triangles

We will use the **Law of Cosines** to find the missing sides in both triangles.

#### (a) Triangle with sides 9 cm, 7 cm, and angle 81°
For this triangle, we have:
- \( a = 9 \, \text{cm} \),
- \( b = 7 \, \text{cm} \),
- \( C = 81^\circ \).

The Law of Cosines states:
\[
c^2 = a^2 + b^2 - 2ab \cdot \cos(C)
\]
Substitute the known values:
\[
c^2 = 9^2 + 7^2 - 2 \cdot 9 \cdot 7 \cdot \cos(81^\circ)
\]
First, compute the cosine of \( 81^\circ \):
\[
\cos(81^\circ) \approx 0.1564
\]
Now, plug this in:
\[
c^2 = 81 + 49 - 2 \cdot 9 \cdot 7 \cdot 0.1564 = 130 - 19.67 \approx 110.33
\]
\[
c \approx \sqrt{110.33} \approx 10.5 \, \text{cm}
\]

So, the missing side is approximately **10.5 cm**.

---

#### (b) Triangle with sides 6 m, 6.8 m, and angle 130°
For this triangle:
- \( a = 6 \, \text{m} \),
- \( b = 6.8 \, \text{m} \),
- \( C = 130^\circ \).

Again, using the Law of Cosines:
\[
c^2 = a^2 + b^2 - 2ab \cdot \cos(C)
\]
Substitute the known values:
\[
c^2 = 6^2 + 6.8^2 - 2 \cdot 6 \cdot 6.8 \cdot \cos(130^\circ)
\]
First, compute the cosine of \( 130^\circ \):
\[
\cos(130^\circ) \approx -0.6428
\]
Now, plug this in:
\[
c^2 = 36 + 46.24 + 2 \cdot 6 \cdot 6.8 \cdot 0.6428
\]
\[
c^2 = 36 + 46.24 + 52.42 = 134.66
\]
\[
c \approx \sqrt{134.66} \approx 11.6 \, \text{m}
\]

So, the missing side is approximately **11.6 m**.

---

Do you have any questions or want more details?

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Here are some related questions:
1. How does the Law of Cosines differ from the Law of Sines?
2. What is another formula for the area of a rhombus?
3. Can the Law of Cosines be applied to right-angled triangles?
4. What would the perimeter of the rhombus be in this case?
5. How does the angle of the rhombus affect its area?

**Tip**: For non-right triangles, always consider using the Law of Cosines when two sides and the included angle are known.

A rhombus has sides of length 5 cm and an angle of 65°. Find its area. Find the length of the remaining side in each triangle (a: 9 cm, 7 cm, 81°; b: 6 m, 6.8 m, 130°).

Learn how to find the area of a rhombus with side lengths of 5 cm and an angle of 65°. This detailed solution also covers how to use the Law of Cosines to find the missing sides of triangles with given sides and angles. Ideal for students in Grades 9-11.

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