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 cm², what is the length (in cm) of its third side?

To solve for the length of the third side in an obtuse-angled triangle \( \triangle ABC \), we can utilize the given information: 

- \( AB = 50 \, \text{cm} \) (longest side),
- One of the other sides \( AC = 42 \, \text{cm} \),
- The area of the triangle is \( 294 \, \text{cm}^2 \).

We need to find the length of the third side \( BC = x \, \text{cm} \).

### Step 1: Area formula
The area of a triangle can also be computed using Heron's formula:

\[
\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}
\]

where \( s \) is the semi-perimeter of the triangle, and \( a \), \( b \), and \( c \) are the side lengths. The semi-perimeter \( s \) is given by:

\[
s = \frac{a + b + c}{2}
\]

We know the area \( \text{Area} = 294 \, \text{cm}^2 \), and two sides \( AB = 50 \, \text{cm} \), \( AC = 42 \, \text{cm} \), so the third side is \( BC = x \, \text{cm} \).

### Step 2: Plugging values
Let's calculate the semi-perimeter \( s \) and then use Heron's formula to find \( x \).The length of the third side \( BC \) is approximately \( 28 \, \text{cm} \).

Would you like more details on how this was calculated, or do you have any other questions?

Here are some related questions for further exploration:

1. How is Heron's formula derived, and why does it work for any triangle?
2. What conditions must be met for a triangle to be classified as obtuse?
3. How can the area of a triangle be calculated using the sine of an angle?
4. Could the cosine rule be used to find the third side of the triangle in this case?
5. How does knowing that the triangle is obtuse affect the choice of solution method?

**Tip:** When solving for unknown sides in triangles, always verify whether trigonometric or geometric formulas, like Heron's formula or the cosine rule, are more efficient for the given problem.

Solve for the third side of an obtuse-angled triangle with sides 50 cm, 42 cm, and area 294 cm² using Heron's formula. This problem explores triangle properties and geometric formulas, suitable for high school students in Grades 9-12.

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