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?

Let the sides of the obtuse-angled triangle be:

- \( AB = c = 50 \, \text{cm} \) (longest side)
- \( AC = a = 42 \, \text{cm} \) (one of the other two sides)
- \( BC = b \) (the side we need to find)

Given the area of the triangle is \( 294 \, \text{cm}^2 \), we can use the formula for the area of a triangle given its sides, known as Heron's formula:

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

where \( s \) is the semi-perimeter of the triangle:

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

Substituting the known values:

\[
s = \frac{42 + b + 50}{2} = \frac{92 + b}{2}
\]

Now, substituting into Heron's formula:

\[
294 = \sqrt{\left(\frac{92 + b}{2}\right) \left(\frac{92 + b}{2} - 42\right) \left(\frac{92 + b}{2} - b\right) \left(\frac{92 + b}{2} - 50\right)}
\]

Simplify each term inside the square root:

1. \( \frac{92 + b}{2} - 42 = \frac{92 + b - 84}{2} = \frac{b + 8}{2} \)
2. \( \frac{92 + b}{2} - b = \frac{92 - b}{2} \)
3. \( \frac{92 + b}{2} - 50 = \frac{92 + b - 100}{2} = \frac{b - 8}{2} \)

Thus, the equation becomes:

\[
294 = \sqrt{\left(\frac{92 + b}{2}\right) \left(\frac{b + 8}{2}\right) \left(\frac{92 - b}{2}\right) \left(\frac{b - 8}{2}\right)}
\]

To simplify this, let’s calculate the values of \( b \).The possible lengths of the third side \( b \) are approximately:

1. \( b \approx 91.08 \, \text{cm} \)
2. \( b \approx 15.23 \, \text{cm} \)

Since the triangle is obtuse-angled with the longest side being \( AB = 50 \, \text{cm} \), the appropriate side length that maintains the triangle's properties is:

\[
b \approx 91.08 \, \text{cm}
\]

Would you like more details on the calculations, or do you have any other questions?

### Related Questions:
1. How does Heron's formula work, and why is it useful in triangle problems?
2. What are the conditions for a triangle to be obtuse-angled?
3. Can you solve the problem using trigonometric methods?
4. How can we verify that a triangle with sides 42 cm, 50 cm, and 91.08 cm is obtuse?
5. What other properties of obtuse triangles are important in geometry?

**Tip:** When working with triangles, always check the triangle inequality theorem to ensure the sides can form a valid triangle.

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?

Find the length of the third side of an obtuse-angled triangle with sides 50 cm and 42 cm, given the area is 294 cm². This problem involves Heron's formula, the semi-perimeter concept, and properties of obtuse-angled triangles. Suitable for students in Grades 9-11.

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