What is an Orthocentre?

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Grade Level:

Class 6

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Definition

What is it?

The orthocentre is a special point inside a triangle. It is where all three 'altitudes' of the triangle meet. An altitude is a line drawn from a vertex (corner) of a triangle perpendicular (at a 90-degree angle) to the opposite side.

Simple Example

Quick Example

Imagine you have a triangular piece of land. If you draw lines from each corner straight down to the opposite boundary, making sure these lines are perfectly upright (like a wall), all three lines will cross at one single point. That meeting point is the orthocentre.

Worked Example

Step-by-Step

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

Step 1: Find the equation of the altitude from A to BC.
--- The side BC lies on the x-axis (y=0). The altitude from A to BC will be a vertical line passing through A(0,4). Its equation is x = 0.
--- Step 2: Find the slope of side AC.
--- Slope of AC = (0 - 4) / (-3 - 0) = -4 / -3 = 4/3.
--- Step 3: Find the slope of the altitude from B to AC.
--- Since the altitude is perpendicular to AC, its slope will be the negative reciprocal of 4/3, which is -3/4.
--- Step 4: Find the equation of the altitude from B to AC.
--- Using point-slope form y - y1 = m(x - x1) with B(3,0) and m = -3/4: y - 0 = (-3/4)(x - 3) => y = (-3/4)x + 9/4.
--- Step 5: Find the intersection point of the two altitudes (x=0 and y = (-3/4)x + 9/4).
--- Substitute x=0 into the second equation: y = (-3/4)(0) + 9/4 => y = 9/4.
--- Step 6: The orthocentre is at (0, 9/4).
--- Answer: The orthocentre of the triangle is (0, 9/4).

Why It Matters

Understanding the orthocentre is important in geometry and has applications in fields like engineering and computer graphics. Engineers use these geometric properties when designing structures like bridges or buildings to ensure stability. In computer science, it helps in creating realistic 3D models and animations.

Common Mistakes

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

MISTAKE: Thinking the orthocentre is always inside the triangle. | CORRECTION: For obtuse triangles (one angle greater than 90 degrees), the orthocentre lies outside the triangle. For right-angled triangles, it lies at the vertex with the right angle.

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

Practice Questions

Try It Yourself

QUESTION: What type of angle must an altitude make with the side it connects to? | ANSWER: 90 degrees (a right angle)

QUESTION: If a triangle has vertices P(1,1), Q(5,1), and R(3,4), find the equation of the altitude from R to PQ. | ANSWER: The line PQ is horizontal (y=1). The altitude from R(3,4) to PQ will be a vertical line passing through R. So, its equation is x = 3.

QUESTION: For a right-angled triangle, where is the orthocentre located? | ANSWER: At the vertex where the right angle is formed.

MCQ

Quick Quiz

Which of the following lines must form a 90-degree angle with the opposite side when drawn from a triangle's vertex?

Median

Angle Bisector

Altitude

Perpendicular Bisector

The Correct Answer Is:

An altitude is defined as a line segment from a vertex perpendicular (at a 90-degree angle) to the opposite side. Medians go to the midpoint, and angle bisectors divide the angle.

Real World Connection

In the Real World

The concept of orthocentre and altitudes is used in many engineering designs. For instance, when architects design the roof trusses of a building, understanding how lines meet at perpendicular angles helps ensure the structure is stable and can withstand forces. It's also used in surveying land, where surveyors use perpendicular measurements to accurately map out triangular plots.

Key Vocabulary

Key Terms

ALTITUDE: A line from a vertex perpendicular to the opposite side | PERPENDICULAR: Forming a 90-degree angle | VERTEX: A corner of a triangle | TRIANGLE: A polygon with three sides and three angles | CONCURRENT: Lines or curves that all intersect at a single common point.

What's Next

What to Learn Next

Great job understanding the orthocentre! Next, you should explore other special points in a triangle like the centroid and the circumcentre. Knowing these will help you appreciate the rich geometry of triangles even more and prepare you for advanced concepts.