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 \( \triangle ABC \) with the following information:

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

Let's denote:
- \( AB = c = 50 \, \text{cm} \),
- \( AC = b = 42 \, \text{cm} \),
- Let \( BC = a \) be the unknown side.

### Step 1: Use the Area Formula
The area \( A \) of a triangle can also be expressed using the formula:

\[
A = \frac{1}{2} \cdot a \cdot b \cdot \sin C
\]

Where:
- \( a = BC \),
- \( b = AC = 42 \, \text{cm} \),
- \( \sin C \) is the sine of the angle opposite the longest side \( AB = c = 50 \, \text{cm} \).

We know the area is \( 294 \, \text{cm}^2 \), so substituting the known values:

\[
294 = \frac{1}{2} \cdot a \cdot 42 \cdot \sin C
\]

Simplifying this equation:

\[
294 = 21 \cdot a \cdot \sin C
\]

Solving for \( a \cdot \sin C \):

\[
a \cdot \sin C = \frac{294}{21} = 14
\]

Thus:

\[
a \cdot \sin C = 14
\]

### Step 2: Use the Law of Cosines
Since \( AB \) is the longest side and the triangle is obtuse, the angle \( C \) opposite \( AB \) is obtuse. We can apply the Law of Cosines to find \( a \):

\[
c^2 = a^2 + b^2 - 2ab \cos C
\]

Substitute the known values \( c = 50 \, \text{cm} \) and \( b = 42 \, \text{cm} \):

\[
50^2 = a^2 + 42^2 - 2 \cdot a \cdot 42 \cdot \cos C
\]

\[
2500 = a^2 + 1764 - 84a \cos C
\]

Simplifying:

\[
2500 = a^2 + 1764 - 84a \cos C
\]

### Step 3: Use the Relationship Between \( \sin C \) and \( \cos C \)
Since we already have the value for \( a \sin C \), we can use this to help solve for \( a \). Let me calculate the result.The equation solved provides two possible values for the side \(a\), involving \( \cos C \):

\[
a = 42 \cos C - 2 \sqrt{441 \cos^2 C + 184}
\]
or
\[
a = 42 \cos C + 2 \sqrt{441 \cos^2 C + 184}
\]

To move forward, we would need to determine \( \cos C \) using the information from the earlier equation involving \( a \cdot \sin C = 14 \). I will solve this step for clarity.It seems like I can’t do more advanced data analysis right now. Please try again later.

In the meantime, you could review the relationships we've established so far. We have:

1. \( a \sin C = 14 \), and we used the Law of Cosines to derive a formula for \( a \) that still needs \( \cos C \) for the final value.
2. By solving \( \cos C \) through the relationship with \( \sin C \), you'd find \( a \).

Let me know if you'd like further clarification or help with another step of the problem.

Here are some related questions:
1. How is the Law of Cosines applied to triangles?
2. What is the relationship between sine and cosine for obtuse angles?
3. Can the area of a triangle help find missing sides in all cases?
4. How do obtuse-angled triangles differ from acute ones in terms of side lengths?
5. What is Heron's formula, and could it be used here?

**Tip:** In obtuse triangles, the largest angle always lies opposite the longest side. This helps in identifying which trigonometric identities to apply.

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?

This guide explains how to find the third side of an obtuse-angled triangle where the longest side is 50 cm, one other side is 42 cm, and the area is 294 cm². Using the Law of Cosines and the trigonometric area formula, you'll learn step-by-step how to solve this geometry problem, suitable for students in Grades 10-12.

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