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 find the length of the third side of an obtuse-angled triangle, we will use the following information:

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

### Step 1: Formula for the Area of a Triangle
The area of a triangle can be expressed using the formula:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]
However, since we are not given the height, we can use **Heron’s formula** to relate the sides of the triangle and its area. First, we need to calculate the third side, which we will call \( BC \), using Heron's formula.

### Step 2: Heron's Formula
Heron’s formula for the area \( A \) of a triangle with sides \( a \), \( b \), and \( c \) is:
\[
A = \sqrt{s(s-a)(s-b)(s-c)}
\]
where \( s \) is the semi-perimeter of the triangle:
\[
s = \frac{a + b + c}{2}
\]
In this case:
- \( a = 50 \, \text{cm} \) (side \( AB \)),
- \( b = 42 \, \text{cm} \) (side \( AC \)),
- \( c = BC \) (unknown).

The area is given as \( 294 \, \text{cm}^2 \).

### Step 3: Solving for \( BC \)
Let's use Heron's formula and solve for \( BC \). I’ll now calculate that.It seems there was an issue while trying to compute the square root in Heron's formula, likely due to unrealistic side lengths in one of the iterations. Let me correct the approach to ensure accurate computation of the third side using better estimation techniques. I’ll retry the calculation.The length of the third side \( BC \) of the triangle is approximately \( 28 \, \text{cm} \).

Would you like further details on how this was calculated or any other related questions?

Here are five questions you could explore next:
1. How does Heron's formula work in detail for triangles?
2. How can we verify if a triangle is obtuse using side lengths?
3. What are the conditions for a valid triangle based on side lengths (triangle inequality)?
4. How can the area of a triangle be computed using trigonometry instead of Heron's formula?
5. What are some real-world applications of finding unknown sides in triangles?

**Tip:** In any triangle, the sum of the lengths of any two sides must be greater than the length of the third side. This is known as the *triangle inequality*.

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?

Learn how to find the third side of an obtuse-angled triangle given its longest side (50 cm), another side (42 cm), and its area (294 cm²). Explore the use of Heron's formula and triangle inequality in this detailed solution. Ideal for students in Grades 9-11.

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