What is Orthocentre? | Simple Explanation for Class 7

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What is the Orthocentre of a Triangle in Coordinate Geometry?

Grade Level:

Class 7

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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. In coordinate geometry, we find this point by using the coordinates of the triangle's vertices and the concept of slopes.

Simple Example

Quick Example

Imagine you have a triangular piece of land in your village, with corners at specific GPS locations. If you draw straight lines from each corner perpendicular to the opposite side (these are altitudes), the point where these three lines cross is the orthocentre. It's like finding a central meeting spot based on heights.

Worked Example

Step-by-Step

Let's find the orthocentre of a triangle with vertices A(1,1), B(5,1), and C(3,5).

1. Find the slope of side BC. Slope m_BC = (5-1)/(3-5) = 4/(-2) = -2.
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2. The altitude from A to BC will be perpendicular to BC. So, its slope m_AD = -1/m_BC = -1/(-2) = 1/2.
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3. The equation of the altitude from A(1,1) with slope 1/2 is y - 1 = (1/2)(x - 1), which simplifies to 2y - 2 = x - 1, or x - 2y = -1. (Equation 1)
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4. Find the slope of side AC. Slope m_AC = (5-1)/(3-1) = 4/2 = 2.
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5. The altitude from B to AC will be perpendicular to AC. So, its slope m_BE = -1/m_AC = -1/2.
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6. The equation of the altitude from B(5,1) with slope -1/2 is y - 1 = (-1/2)(x - 5), which simplifies to 2y - 2 = -x + 5, or x + 2y = 7. (Equation 2)
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7. Now, solve Equation 1 and Equation 2 simultaneously to find the intersection point (the orthocentre).
(x - 2y = -1) + (x + 2y = 7)
2x = 6 => x = 3
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8. Substitute x = 3 into Equation 2: 3 + 2y = 7 => 2y = 4 => y = 2.
So, the orthocentre is (3,2).

Why It Matters

Understanding the orthocentre is crucial in fields like engineering for designing stable structures or in computer graphics for precise object placement. Architects use these geometric principles to ensure buildings are balanced and strong. It's also foundational for advanced math in data science and physics.

Common Mistakes

MISTAKE: Confusing altitude with median or angle bisector. | CORRECTION: An altitude is a line segment from a vertex perpendicular to the opposite side. A median goes to the midpoint, and an angle bisector divides the angle.

MISTAKE: Incorrectly calculating the slope of the perpendicular line. | CORRECTION: If the slope of a line is 'm', the slope of a perpendicular line is '-1/m'. Remember to flip the fraction and change the sign.

MISTAKE: Making calculation errors when solving simultaneous equations. | CORRECTION: Double-check each step when solving the two altitude equations. Use elimination or substitution carefully to find the correct (x,y) coordinates.

Practice Questions

Try It Yourself

QUESTION: What is the slope of an altitude if the slope of the side it is perpendicular to is 3? | ANSWER: -1/3

QUESTION: Find the equation of the altitude from vertex P(2,3) to side QR, where QR has a slope of -1. | ANSWER: y - 3 = 1(x - 2) or y = x + 1

QUESTION: A triangle has vertices A(0,0), B(4,0), and C(2,6). Find the orthocentre of this triangle. | ANSWER: (2, 2/3)

MCQ

Quick Quiz

Which of these lines passes through the orthocentre of a triangle?

Median

Angle Bisector

Altitude

Perpendicular Bisector

The Correct Answer Is:

The orthocentre is defined as the intersection point of the altitudes. Medians intersect at the centroid, angle bisectors at the incenter, and perpendicular bisectors at the circumcenter.

Real World Connection

In the Real World

In cricket, when analysts use software to track the path of a ball, they often use coordinate geometry. Imagine the path of a cricket ball forming a triangle with points on the ground. Understanding altitudes and orthocentres can help in calculating precise angles and trajectories for bowling machines or for predicting ball bounce, just like how ISRO scientists use precise geometry for satellite orbits.

Key Vocabulary

Key Terms

ALTITUDE: A line segment from a vertex perpendicular to the opposite side. | PERPENDICULAR: Forming a 90-degree angle. | SLOPE: A measure of the steepness of a line. | COORDINATE GEOMETRY: Using coordinates to study geometric figures. | VERTEX: A corner point of a triangle.

What's Next

What to Learn Next

Great job learning about the orthocentre! Next, you can explore the 'Centroid of a Triangle' and the 'Circumcentre of a Triangle'. These are other special points within a triangle, and understanding them will deepen your knowledge of triangle properties and coordinate geometry.