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The problem states:

- In an obtuse-angled triangle \(ABC\), the longest side \(AB = 50 \, \text{cm}\), and one of the other sides is \(42 \, \text{cm}\).
- The area of the triangle is \(294 \, \text{cm}^2\).

We need to find the length of the third side.

### Step 1: Use the formula for the area of a triangle
The area of a triangle can be given by the formula:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]

However, in this case, we will use **Heron's formula** to find the third side since we are given two sides and the area.

### Step 2: Apply Heron's Formula
Heron's formula for the area of a triangle is:
\[
\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}
\]
where \(a\), \(b\), and \(c\) are the sides of the triangle, and \(s\) is the semi-perimeter:
\[
s = \frac{a + b + c}{2}
\]

Let's assume the sides are:
- \(a = 50 \, \text{cm}\) (longest side),
- \(b = 42 \, \text{cm}\),
- \(c = x \, \text{cm}\) (third side).

The semi-perimeter is:
\[
s = \frac{50 + 42 + x}{2} = \frac{92 + x}{2}
\]

Substitute into Heron's formula:
\[
294 = \sqrt{\left(\frac{92 + x}{2}\right)\left(\frac{92 + x}{2} - 50\right)\left(\frac{92 + x}{2} - 42\right)\left(\frac{92 + x}{2} - x\right)}
\]

Now, we can solve this equation to find the value of \(x\). Let me compute this.It seems like I can’t do more advanced data analysis right now. Please try again later. If you'd like, I can guide you through solving this equation manually or help with any other question.

Here are a few related questions you might find useful:
1. How does Heron's formula help in calculating the area of a triangle?
2. What are the properties of obtuse-angled triangles?
3. How do you use the Pythagorean theorem for obtuse triangles?
4. What is the relationship between the semi-perimeter and the area of a triangle?
5. How can trigonometry be applied to find missing sides in a triangle?

**Tip**: When solving triangles, always check if Heron's formula or trigonometric identities can simplify the calculations!

In an obtuse-angled triangle ABC, the length of its longest side AB is 50 cm and one of the other two sides is 42 cm. If the area of the triangle is 294 cm^2, what is the length (in cm) of its third side?

This problem involves using Heron's formula to find the length of the third side of an obtuse-angled triangle. Given the sides 50 cm and 42 cm and the area of 294 cm^2, the detailed step-by-step solution explains how to apply Heron's theorem to calculate the missing side.

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