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

We are given an obtuse-angled triangle \( ABC \) where:

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

We need to find the length of the third side \( BC \), denoted by \( x \).

### Step 1: Use the formula for the area of a triangle

The area of a triangle can also be expressed as:

\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]

However, we don't have the height, so we'll use **Heron's formula** to solve this problem.

### Step 2: Apply Heron's formula

The area of a triangle with sides \( a \), \( b \), and \( c \) can be given by Heron's formula:

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

where \( s \) is the semi-perimeter of the triangle, and is given by:

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

Here, \( a = 50 \, \text{cm}, b = 42 \, \text{cm}, \text{ and } c = x \). 

Thus, the semi-perimeter is:

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

### Step 3: Set up the equation using Heron's formula

From the problem, we know the area is 294 cm². Substituting into Heron’s formula:

\[
294 = \sqrt{s(s - 50)(s - 42)(s - x)}
\]

We now solve this equation to find \( x \). I will calculate it step by step.The solutions for the third side \( x \) are:

\[
x = 2\sqrt{2074}, \quad x = -2\sqrt{2074}, \quad x = 2\sqrt{58}, \quad x = -2\sqrt{58}
\]

Since \( x \) must be positive, the valid solutions are \( x = 2\sqrt{2074} \) and \( x = 2\sqrt{58} \). Let me now evaluate the numerical values of these solutions.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, you can manually evaluate the possible lengths of the third side \( x \):

- \( x = 2\sqrt{2074} \approx 91.08 \, \text{cm} \)
- \( x = 2\sqrt{58} \approx 15.23 \, \text{cm} \)

Since the triangle is obtuse and \( AB = 50 \, \text{cm} \) is the longest side, the third side must be smaller than 50 cm. Therefore, \( x = 15.23 \, \text{cm} \) is the correct length of the third side.

Would you like further clarification or details?

Here are 5 related questions for you to explore:
1. How does Heron’s formula work for different types of triangles?
2. How do you identify the obtuse angle in a triangle?
3. What are the conditions for a triangle to be obtuse?
4. How does the Pythagorean theorem relate to obtuse triangles?
5. Can you calculate the height of the triangle if the base and area are known?

**Tip:** In obtuse triangles, the longest side is always opposite the obtuse angle.

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 calculate the length of the third side in an obtuse-angled triangle using Heron's formula. This problem involves a triangle with sides 50 cm, 42 cm, and an area of 294 cm². Step-by-step solution included. Suitable for students in Grades 9-11.

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