Vectors - Problems - Proving triangle is equilateral

• In this topic, we will learn how to prove that a triangle is equilateral using vector methods.
• We will explore different cases and techniques to solve such problems.
• Let’s get started!

Case 1: Proving triangle is equilateral

• Given: Triangle ABC
• To prove: Triangle ABC is equilateral
• Approach: We will use vector methods to solve this problem.

Approach

1. Calculate the vectors AB, BC, and CA.

2. Find the magnitudes of vectors AB, BC, and CA.

3. If the magnitudes are equal, the triangle is equilateral.

4. If the magnitudes are not equal, the triangle is not equilateral.

Example 1

Given:

• A(-1, 2)
• B(3, -4)
• C(7, 2) To prove: Triangle ABC is equilateral

Solution 1

1. Vector AB = Vector B - Vector A = (3, -4) - (-1, 2) = (4, -6)

2. Vector BC = Vector C - Vector B = (7, 2) - (3, -4) = (4, 6)

3. Vector CA = Vector A - Vector C = (-1, 2) - (7, 2) = (-8, 0)

4. Magnitude of AB = √(4² + (-6)²) = √(16 + 36) = √52

5. Magnitude of BC = √(4² + 6²) = √(16 + 36) = √52

6. Magnitude of CA = √((-8)² + 0²) = √(64 + 0) = √64 = 8

Solution 2

• Notice that the magnitudes of vectors AB and BC are equal (√52 = √52).
• However, the magnitude of vector CA is not equal to the magnitudes of AB and BC (8 ≠ √52).
• Hence, the triangle ABC is not equilateral.

Case 2: Proving triangle is equilateral

• Given: Triangle ABC
• To prove: Triangle ABC is equilateral
• Approach: We will use the concept of dot products and magnitude.

Approach

1. Calculate vectors AB, AC, and BC.

2. Calculate the dot product of vectors AB and AC, BC and AB, and AC and BC.

3. If the dot products are equal, the triangle is equilateral.

4. If the dot products are not equal, the triangle is not equilateral.

Example 2

Given:

• A(1, 3)
• B(2, 6)
• C(4, 3) To prove: Triangle ABC is equilateral

Solution 3

1. Vector AB = Vector B - Vector A = (2, 6) - (1, 3) = (1, 3)

2. Vector AC = Vector C - Vector A = (4, 3) - (1, 3) = (3, 0)

3. Vector BC = Vector C - Vector B = (4, 3) - (2, 6) = (2, -3)

4. Dot product of AB and AC = (1, 3) • (3, 0) = 1 * 3 + 3 * 0 = 3

5. Dot product of BC and AB = (2, -3) • (1, 3) = 2 * 1 + (-3) * 3 = -7

6. Dot product of AC and BC = (3, 0) • (2, -3) = 3 * 2 + 0 * (-3) = 6

Solution 4

• The dot products of vectors AB and AC, BC and AB, and AC and BC are not equal (3 ≠ -7 ≠ 6).
• Hence, the triangle ABC is not equilateral.

Case 3: Proving triangle is equilateral

• Given: Triangle ABC
• To prove: Triangle ABC is equilateral
• Approach: We will use the concept of angle between vectors.

Approach

1. Calculate vectors AB, AC, and BC.

2. Calculate the angles between vectors AB and AC, BC and AB, and AC and BC.

3. If the angles are equal, the triangle is equilateral.

4. If the angles are not equal, the triangle is not equilateral.

Example 3

Given:

• A(2, 3)
• B(4, 1)
• C(6, 3) To prove: Triangle ABC is equilateral

Solution 5

1. Vector AB = Vector B - Vector A = (4, 1) - (2, 3) = (2, -2)

2. Vector AC = Vector C - Vector A = (6, 3) - (2, 3) = (4, 0)

3. Vector BC = Vector C - Vector B = (6, 3) - (4, 1) = (2, 2)

4. Angle between AB and AC = arccos((2, -2) • (4, 0) / (√(2² + (-2)²) * √(4² + 0²))

5. Angle between BC and AB = arccos((2, 2) • (2, -2) / (√(2² + 2²) * √(2² + (-2)²))

6. Angle between AC and BC = arccos((4, 0) • (2, 2) / (√(4² + 0²) * √(2² + 2²))

Solution 6

• The angles between vectors AB and AC, BC and AB, and AC and BC are not equal.
• Hence, the triangle ABC is not equilateral.

Case 4: Proving triangle is equilateral

• Given: Triangle ABC
• To prove: Triangle ABC is equilateral
• Approach: We will use the concept of vector components.

Approach

1. Calculate vectors AB, AC, and BC.

2. Calculate the components of vectors AB, AC, and BC along the x-axis and y-axis.

3. If the components are equal, the triangle is equilateral.

4. If the components are not equal, the triangle is not equilateral.

Example 4

Given:

• A(1, 1)
• B(2, 4)
• C(3, 1) To prove: Triangle ABC is equilateral

Solution 7

1. Vector AB = Vector B - Vector A = (2, 4) - (1, 1) = (1, 3)

2. Vector AC = Vector C - Vector A = (3, 1) - (1, 1) = (2, 0)

3. Vector BC = Vector C - Vector B = (3, 1) - (2, 4) = (1, -3)

4. Components of AB along the x-axis = 1

5. Components of AC along the x-axis = 2

6. Components of BC along the x-axis = 1

Solution 8

• The components of vectors AB, AC, and BC along the x-axis are not equal.
• Hence, the triangle ABC is not equilateral.

Summary

• We explored different techniques to prove that a triangle is equilateral using vector methods.
• We used the concepts of magnitudes, dot products, angles between vectors, and vector components.
• By applying these techniques, we were able to determine whether a given triangle is equilateral or not.

Key Points to Remember

1. To prove a triangle is equilateral, we need to show that the magnitudes of its sides are equal.

2. We can also use the dot products of vectors to determine if a triangle is equilateral.

3. The angles between vectors can also be used to prove the equilateral nature of a triangle.

4. Vector components along the x-axis and y-axis can help us determine if a triangle is equilateral.

Practice Problems

1. Given: A(2, 2), B(4, 2), C(3, 4). Prove that triangle ABC is equilateral.

2. Given: A(-3, 1), B(-1, -1), C(1, 3). Prove that triangle ABC is equilateral.

3. Given: A(5, 2), B(6, 4), C(7, 2). Prove that triangle ABC is equilateral.

Practice Problem Solutions

1. Solution: AB = (4 - 2, 2 - 2) = (2, 0); BC = (3 - 4, 4 - 2) = (-1, 2); AC = (3 - 2, 4 - 2) = (1, 2). Magnitude of AB = 2; magnitude of BC = √5; magnitude of AC = √5. Hence, the triangle ABC is not equilateral.

2. Solution: AB = (-1 - (-3), -1 - 1) = (2, -2); BC = (1 - (-1), 3 - (-1)) = (2, 4); AC = (1 - (-3), 3 - 1) = (4, 2). Magnitude of AB = √8; magnitude of BC = √20; magnitude of AC = √20. Hence, the triangle ABC is equilateral.

3. Solution: AB = (6 - 5, 4 - 2) = (1, 2); BC = (7 - 6, 2 - 4) = (1, -2); AC = (7 - 5, 2 - 2) = (2, 0). Magnitude of AB = √5; magnitude of BC = √5; magnitude of AC = 2. Hence, the triangle ABC is not equilateral.