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

In this problem, we are given an obtuse-angled triangle \(ABC\) where the longest side \(AB = 50\) cm, one of the other two sides \(AC = 42\) cm, and the area of the triangle is 294 cm². We need to find the length of the third side \(BC\).

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

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

However, since we are not given the height, we can use **Heron's formula** for the area of a triangle when all sides are known.

### Step 2: Heron's Formula
The area \(A\) of a triangle with sides \(a\), \(b\), and \(c\) is given by:

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

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

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

In our case, the sides are:
- \(AB = 50\) cm,
- \(AC = 42\) cm,
- Let \(BC = x\) cm (the side we are trying to find).

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

The area is given as 294 cm², so:

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

Now, we need to solve for \(x\). Let's first compute the semi-perimeter \(s\) in terms of \(x\):

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

Substitute this into the Heron's formula and solve for \(x\). I’ll now calculate the result.The solutions for the length of the third side \(BC\) (denoted as \(x\)) are:

\[
x = 2\sqrt{2074} \quad \text{or} \quad x = 2\sqrt{58}
\]

Since the side length of a triangle cannot be negative, we reject the negative solutions. The positive values are approximately:

\[
x \approx 2\sqrt{2074} \quad \text{or} \quad 2\sqrt{58}
\]

Let's now compute the numerical values for these solutions.It seems like I can’t do more advanced data analysis right now. Please try again later. However, I can explain the approximate values for the solutions:

- \( 2\sqrt{2074} \) is approximately 91 cm
- \( 2\sqrt{58} \) is approximately 15 cm

Since \(AB = 50\) cm is the longest side, the length of the third side \(BC\) must be around 91 cm (as the smaller solution would violate the longest side condition).

Let me know if you'd like further details or have any other questions!

Here are 5 related questions for practice:
1. How would you use Heron's formula if only the area and two sides were given?
2. What changes when dealing with obtuse-angled triangles compared to acute triangles?
3. How can the sine rule help in solving triangle problems with known angles?
4. How would the problem change if the triangle were a right triangle?
5. How do we know which side is the longest in obtuse-angled triangles?

**Tip**: For obtuse-angled triangles, the angle opposite the longest side is greater than 90°. This can help in identifying unknown sides or angles.

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?

Solve the problem of finding the third side of an obtuse-angled triangle with sides 50 cm and 42 cm and an area of 294 cm². This solution covers Heron's formula for the area of a triangle and the method to find the third side. Suitable for Grades 10-12 students.

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