What is the Orthocentre of a Triangle? | Simple Explanation for Class 6

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What is the Orthocentre of a Triangle (Coordinate Geometry)?

Grade Level:

Class 6

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Definition

What is it?

The orthocentre of a triangle is a special point where all three altitudes of the triangle meet. An altitude is a line segment drawn from a vertex of the triangle perpendicular to the opposite side. In coordinate geometry, we find this point using the equations of these perpendicular lines.

Simple Example

Quick Example

Imagine you have a triangular piece of land in your village. If you draw a straight line from each corner to the opposite boundary, making sure the line forms a perfect 'L' shape (90 degrees) with the boundary, all three lines will cross at one single point. That crossing point is the orthocentre.

Worked Example

Step-by-Step

Let's find the orthocentre of a triangle with vertices A(0, 0), B(6, 0), and C(3, 4).

Step 1: Find the slope of side BC. Slope m_BC = (4 - 0) / (3 - 6) = 4 / -3 = -4/3.
---Step 2: The altitude from A to BC will be perpendicular to BC. So, its slope m_AD = -1 / m_BC = -1 / (-4/3) = 3/4.
---Step 3: The equation of the altitude AD (passing through A(0,0) with slope 3/4) is y - 0 = (3/4)(x - 0), which simplifies to y = (3/4)x or 4y = 3x.
---Step 4: Find the slope of side AC. Slope m_AC = (4 - 0) / (3 - 0) = 4/3.
---Step 5: The altitude from B to AC will be perpendicular to AC. So, its slope m_BE = -1 / m_AC = -1 / (4/3) = -3/4.
---Step 6: The equation of the altitude BE (passing through B(6,0) with slope -3/4) is y - 0 = (-3/4)(x - 6), which simplifies to y = (-3/4)x + 18/4 or 4y = -3x + 18.
---Step 7: Now, we solve the two altitude equations to find their intersection point (the orthocentre). We have 4y = 3x and 4y = -3x + 18. Since both are equal to 4y, we can set them equal to each other: 3x = -3x + 18.
---Step 8: Solve for x: 6x = 18, so x = 3. Substitute x = 3 into 4y = 3x: 4y = 3(3), so 4y = 9, which means y = 9/4.

Answer: The orthocentre of the triangle is (3, 9/4).

Why It Matters

Understanding the orthocentre helps in fields like engineering to design stable structures or in computer graphics to render realistic 3D objects. Architects use these geometric principles to ensure buildings are balanced, and even game developers use them for character movements and object placement.

Common Mistakes

MISTAKE: Confusing altitude with median or angle bisector. | CORRECTION: Remember, an altitude MUST be perpendicular to the opposite side (form a 90-degree angle), unlike a median which goes to the midpoint, or an angle bisector which divides the angle.

MISTAKE: Incorrectly calculating the slope of the perpendicular line. | CORRECTION: If a line has slope 'm', the perpendicular line will have a slope of '-1/m'. Don't forget to flip the fraction and change the sign!

MISTAKE: Making calculation errors when solving the system of equations for the intersection point. | CORRECTION: Double-check your arithmetic when substituting values and solving for x and y. A small mistake can lead to a completely wrong orthocentre.

Practice Questions

Try It Yourself

QUESTION: What is the orthocentre of a right-angled triangle with vertices at (0,0), (5,0), and (0,3)? | ANSWER: (0,0)

QUESTION: Find the orthocentre of a triangle with vertices P(1, 2), Q(5, 2), and R(3, 6). | ANSWER: (3, 3)

QUESTION: For a triangle with vertices A(-2, 1), B(4, 1), and C(1, 5), find the equations of two altitudes and then determine the orthocentre. | ANSWER: Altitude from C to AB: x = 1. Altitude from B to AC: y - 1 = (3/4)(x - 4). Orthocentre: (1, -5/4)

MCQ

Quick Quiz

Which of the following describes an altitude of a triangle?

A line segment from a vertex to the midpoint of the opposite side.

A line segment that divides a vertex angle into two equal parts.

A line segment from a vertex perpendicular to the opposite side.

A line segment connecting the midpoints of two sides.

The Correct Answer Is:

An altitude is defined as a line segment from a vertex that is perpendicular to the opposite side. Options A, B, and D describe medians, angle bisectors, and midsegments, respectively, which are different concepts.

Real World Connection

In the Real World

Imagine you're an engineer designing a new bridge. To ensure it stands strong, you might use triangles in its structure. Calculating the orthocentre can help determine key stress points or balance points in these triangular supports, ensuring the bridge can handle heavy traffic like trucks carrying goods across India.

Key Vocabulary

Key Terms

ALTITUDE: A line segment from a vertex perpendicular to the opposite side | PERPENDICULAR: Lines or segments that meet at a 90-degree angle | SLOPE: A measure of the steepness of a line | VERTEX: A corner point of a triangle | COORDINATE GEOMETRY: Using coordinates (x, y) to study geometric shapes.

What's Next

What to Learn Next

Great job learning about the orthocentre! Next, you can explore other special points in a triangle like the centroid and circumcentre. Understanding these will give you a complete picture of the 'centres' of a triangle and how they are used in real-world problems.