. Find all the missing parts in the following triangles, ABC

a = 10m

b =7.5.m

c = 17m

Given the triangle \(ABC\) with sides \(a = 10\) m, \(b = 7.5\) m, and \(c = 17\) m, we need to find the missing angles \(A\), \(B\), and \(C\).

To determine these angles, we can use the **Law of Cosines**, which states:

\[
c^2 = a^2 + b^2 - 2ab \cdot \cos(C)
\]

Substituting the known values:

\[
17^2 = 10^2 + 7.5^2 - 2 \cdot 10 \cdot 7.5 \cdot \cos(C)
\]

Now, solve for \(\cos(C)\):

\[
289 = 100 + 56.25 - 150 \cdot \cos(C)
\]

\[
289 = 156.25 - 150 \cdot \cos(C)
\]

\[
132.75 = -150 \cdot \cos(C)
\]

\[
\cos(C) = -\frac{132.75}{150}
\]

\[
\cos(C) = -0.885
\]

Now, find \(C\):

\[
C = \cos^{-1}(-0.885) \approx 152.2^\circ
\]

Since \(C\) is greater than \(90^\circ\), this triangle is obtuse.

Now, use the **Law of Sines** to find the other angles:

\[
\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}
\]

For angle \(A\):

\[
\frac{\sin A}{10} = \frac{\sin 152.2^\circ}{17}
\]

\[
\sin A = \frac{10 \cdot \sin 152.2^\circ}{17}
\]

Calculate \(\sin 152.2^\circ\):

\[
\sin 152.2^\circ \approx 0.437
\]

\[
\sin A = \frac{10 \cdot 0.437}{17} \approx 0.257
\]

\[
A = \sin^{-1}(0.257) \approx 14.9^\circ
\]

For angle \(B\):

\[
B = 180^\circ - C - A = 180^\circ - 152.2^\circ - 14.9^\circ = 12.9^\circ
\]

### Summary of the results:
- \(A \approx 14.9^\circ\)
- \(B \approx 12.9^\circ\)
- \(C \approx 152.2^\circ\)

Do you want more details or have any questions?

---

1. What if \(c\) was smaller than \(a + b\)?
2. How does the Law of Cosines differ from the Law of Sines?
3. Can you apply the Law of Sines directly without knowing one angle?
4. What would change if all angles were less than \(90^\circ\)?
5. How can you verify the results?

**Tip:** For obtuse triangles, the largest angle is opposite the longest side.

Explore how to find missing angles in triangle ABC with given side lengths using the Law of Cosines and Law of Sines. This problem involves determining angles A, B, and C when sides a = 10m, b = 7.5m, and c = 17m. Suitable for Grades 10-12 students studying geometry and trigonometry.

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