In ΔHIJ, h = 48 cm, i = 49 cm and ∠J=171°. Find the area of ΔHIJ, to the nearest square centimeter.

To find the area of triangle \( \triangle HIJ \), where \( h = 48 \ \text{cm} \), \( i = 49 \ \text{cm} \), and \( \angle J = 171^\circ \), we can use the formula involving two sides and the included angle:

\[
\text{Area} = \frac{1}{2} \cdot h \cdot i \cdot \sin(\angle J)
\]

### Step 1: Plug in the known values.
\[
h = 48 \ \text{cm}, \ i = 49 \ \text{cm}, \ \angle J = 171^\circ
\]

\[
\text{Area} = \frac{1}{2} \cdot 48 \cdot 49 \cdot \sin(171^\circ)
\]

### Step 2: Calculate \(\sin(171^\circ)\).
Since \(\sin(171^\circ) = \sin(180^\circ - 171^\circ) = \sin(9^\circ)\):

\[
\sin(9^\circ) \approx 0.1564
\]

### Step 3: Substitute \(\sin(171^\circ)\) into the formula.
\[
\text{Area} = \frac{1}{2} \cdot 48 \cdot 49 \cdot 0.1564
\]

\[
\text{Area} = 24 \cdot 49 \cdot 0.1564 \approx 183.1776
\]

### Step 4: Round to the nearest square centimeter.
\[
\text{Area} \approx 183 \ \text{cm}^2
\]

### Final Answer:
The area of \( \triangle HIJ \) is approximately **183 square centimeters**.

Would you like more details or have any questions?

### Related Questions:
1. How is the sine function used in solving triangle problems?
2. What happens if \(\angle J\) is obtuse in a triangle area calculation?
3. How can the Law of Cosines be used to find unknown sides?
4. What is the formula for the area of a triangle using three sides?
5. How does the angle's measure affect the triangle's area?

### Tip:
Remember, when using trigonometric functions, angles close to \(180^\circ\) can produce small values for the sine function, affecting the area calculations.

Learn how to find the area of triangle ΔHIJ, where h = 48 cm, i = 49 cm, and ∠J = 171°. This detailed guide covers trigonometric calculations, including using the sine rule to find the area of a triangle. Suitable for high school students in Grades 10-12.

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