Finding the missing side of a triangle depends entirely on what information you already have. Triangles have three sides and three angles, and knowing certain combinations allows you to calculate the others using various geometric theorems. This guide will walk you through the most common scenarios.
Understanding the Key Triangle Theorems
Before we dive into solving for missing sides, let's review the essential theorems:
1. Pythagorean Theorem
This theorem applies only to right-angled triangles (triangles with one 90-degree angle). It states:
a² + b² = c²
Where:
- a and b are the lengths of the two shorter sides (legs)
- c is the length of the longest side (hypotenuse), opposite the right angle.
If you know two sides of a right-angled triangle, you can easily find the third using this formula.
2. Sine Rule
The sine rule works for any triangle, not just right-angled ones. It relates the sides and angles of a triangle:
a/sin A = b/sin B = c/sin C
Where:
- a, b, c are the lengths of the sides
- A, B, C are the angles opposite those sides respectively.
If you know two angles and one side, or two sides and one angle (not the angle between the two sides), you can use the sine rule to find a missing side.
3. Cosine Rule
Similar to the sine rule, the cosine rule applies to any triangle. It's particularly useful when you know two sides and the angle between them:
c² = a² + b² - 2ab cos C
Where:
- a, b, c are the lengths of the sides
- C is the angle between sides a and b.
How to Find the Missing Side: Step-by-Step Examples
Let's work through some examples to illustrate how to use these theorems:
Example 1: Right-Angled Triangle
A right-angled triangle has sides of length 3cm and 4cm. Find the length of the hypotenuse.
- Identify the knowns: a = 3cm, b = 4cm.
- Apply the Pythagorean Theorem: c² = a² + b² = 3² + 4² = 9 + 16 = 25
- Solve for c: c = √25 = 5cm
Therefore, the length of the hypotenuse is 5cm.
Example 2: Using the Sine Rule
A triangle has angles A = 30°, B = 70°, and side a = 5cm. Find the length of side b.
- Find angle C: The angles in a triangle add up to 180°, so C = 180° - 30° - 70° = 80°
- Apply the Sine Rule: a/sin A = b/sin B
- Substitute the known values: 5/sin 30° = b/sin 70°
- Solve for b: b = (5 * sin 70°) / sin 30° ≈ 9.4cm
Example 3: Using the Cosine Rule
A triangle has sides a = 6cm, b = 8cm, and the angle between them, C = 60°. Find the length of side c.
- Apply the Cosine Rule: c² = a² + b² - 2ab cos C
- Substitute the known values: c² = 6² + 8² - 2 * 6 * 8 * cos 60° = 36 + 64 - 48 = 52
- Solve for c: c = √52 ≈ 7.2cm
Tips and Tricks for Success
- Draw a diagram: Always start by sketching the triangle. This helps visualize the problem and identify which theorem to use.
- Label your sides and angles: Clearly label each side and angle to avoid confusion.
- Use a calculator: Accurate calculations are crucial, especially when dealing with angles and trigonometric functions.
- Check your answer: Make sure your answer makes sense in the context of the problem.
By understanding these theorems and following these steps, you'll be well-equipped to find the missing side of any triangle, regardless of the information you're given. Remember to practice regularly to build your skills and confidence!
