In triangle PST PS = 7cm ST =9cm and PT = 13cm.  What is the biggest angle? ,

To determine the largest angle in triangle \( PST \), we can use the **Cosine Rule**. The Cosine Rule states:

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

Where:
- \( c \) is the side opposite the angle \( C \),
- \( a \) and \( b \) are the other two sides of the triangle.

For triangle \( PST \):
- \( PS = 7 \) cm,
- \( ST = 9 \) cm,
- \( PT = 13 \) cm.

To find the largest angle, we'll first identify which side is the longest, as the largest angle will be opposite the longest side. Here, \( PT = 13 \) cm is the longest side, so the largest angle is \( \angle S \) opposite side \( PT \).

Now, apply the Cosine Rule to find \( \angle S \):

\[
\cos(S) = \frac{PS^2 + ST^2 - PT^2}{2 \cdot PS \cdot ST}
\]

Substitute the given values:

\[
\cos(S) = \frac{7^2 + 9^2 - 13^2}{2 \cdot 7 \cdot 9}
\]
\[
\cos(S) = \frac{49 + 81 - 169}{126}
\]
\[
\cos(S) = \frac{130 - 169}{126} = \frac{-39}{126}
\]
\[
\cos(S) = -\frac{39}{126} \approx -0.3095
\]

Now, find \( \angle S \) by taking the inverse cosine (arccos):

\[
\angle S = \cos^{-1}(-0.3095) \approx 108^\circ
\]

So, the largest angle \( \angle S \) is approximately \( 108^\circ \).

Would you like more details on this calculation or have any other questions?

Here are some related questions you might find interesting:
1. How do you derive the Cosine Rule from the Pythagorean theorem?
2. What are the properties of a triangle with one angle greater than 90 degrees?
3. How can you find the area of triangle \( PST \) using the side lengths?
4. What is the Sine Rule, and when is it used instead of the Cosine Rule?
5. How would you solve for the other angles in triangle \( PST \)?

**Tip:** Always remember that the largest angle in a triangle is opposite the longest side.

Learn how to determine the largest angle in triangle PST with side lengths PS = 7cm, ST = 9cm, and PT = 13cm using the Cosine Rule. Understand the method of identifying the largest angle opposite the longest side in a triangle. Suitable for Grades 10-12 students studying geometry and trigonometry.

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