In ΔIJK, j = 340 cm, k = 330 cm and ∠I=105°. Find the length of i, to the nearest centimeter.

To find the length of side \(i\) in \(\Delta IJK\), we can use the **Law of Cosines**. The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. It is given by:

\[
i^2 = j^2 + k^2 - 2jk \cdot \cos(\angle I)
\]

where:
- \(i\) is the side opposite \(\angle I\),
- \(j = 340 \, \text{cm}\),
- \(k = 330 \, \text{cm}\),
- \(\angle I = 105^\circ\).

### Step 1: Apply the values into the Law of Cosines formula

\[
i^2 = 340^2 + 330^2 - 2(340)(330) \cdot \cos(105^\circ)
\]

We will now compute the right-hand side to find \(i\).

Let me calculate that for you.The length of side \(i\) in \(\Delta IJK\) is approximately **532 cm**.

Would you like more details or have any questions? Here are five related questions you might find useful:

1. How does the Law of Cosines apply to non-right triangles?
2. How do we convert angles from degrees to radians?
3. Can the Law of Cosines be used for right triangles?
4. What is the relationship between the Law of Cosines and the Law of Sines?
5. How do you interpret obtuse angles like \(105^\circ\) when applying trigonometric formulas?

**Tip:** The Law of Cosines is especially useful for triangles where two sides and the included angle are known, or when all three sides are known but no angles are.

In ΔIJK, j = 340 cm, k = 330 cm and ∠I = 105°. Find the length of i, to the nearest centimeter.

Discover how to calculate the length of side i in ΔIJK using the Law of Cosines. With j = 340 cm, k = 330 cm, and ∠I = 105°, this solution walks through applying trigonometric formulas to find the side length. Ideal for high school students in Grades 10-12.

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