What is the Equation of the Altitude? | Simple Explanation for Class 10

S6-SA1-0310

What is the Equation of the Altitude of a Triangle?

Grade Level:

Class 10

AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine

Definition

What is it?

The equation of the altitude of a triangle is the straight line equation that represents a line segment drawn from one vertex of the triangle perpendicular to the opposite side. This line shows the 'height' of the triangle from that vertex. We find it using the concept of perpendicular lines and their slopes.

Simple Example

Quick Example

Imagine you have a triangular plot of land. If you want to build a straight fence from one corner directly to the opposite boundary such that it makes a perfect 90-degree angle with that boundary, the line representing this fence is an altitude. The equation of this line tells you its exact path on a map.

Worked Example

Step-by-Step

Let's find the equation of the altitude from vertex A(1, 5) to side BC of a triangle with vertices A(1, 5), B(3, 1), and C(7, 3).

Step 1: Find the slope of side BC.
Slope of BC (m_BC) = (y2 - y1) / (x2 - x1) = (3 - 1) / (7 - 3) = 2 / 4 = 1/2.

---Step 2: Find the slope of the altitude from A to BC.
Since the altitude is perpendicular to BC, its slope (m_alt) will be the negative reciprocal of m_BC.
m_alt = -1 / m_BC = -1 / (1/2) = -2.

---Step 3: Use the point-slope form to find the equation of the altitude.
The altitude passes through point A(1, 5) and has a slope of -2.
Point-slope form: y - y1 = m(x - x1)
y - 5 = -2(x - 1)

---Step 4: Simplify the equation to the standard form (Ax + By + C = 0).
y - 5 = -2x + 2
2x + y - 5 - 2 = 0
2x + y - 7 = 0.

---Answer: The equation of the altitude from vertex A to side BC is 2x + y - 7 = 0.

Why It Matters

Understanding altitudes helps engineers design stable structures like bridges and buildings, ensuring they can withstand forces. In robotics, it's used to calculate precise movements and avoid collisions. Even in medical imaging, algorithms use geometric properties like altitudes to analyze shapes of organs or tumors, helping doctors make diagnoses.

Common Mistakes

MISTAKE: Using the same slope for the altitude as the side it's perpendicular to. | CORRECTION: Remember that perpendicular lines have slopes that are negative reciprocals of each other (m1 * m2 = -1).

MISTAKE: Using a point on the opposite side (like B or C) instead of the vertex (A) for the point-slope form. | CORRECTION: The altitude always passes through the vertex from which it is drawn. Use the coordinates of that vertex in the point-slope form.

MISTAKE: Calculation errors in finding the slope or simplifying the equation. | CORRECTION: Double-check your arithmetic, especially when dealing with fractions and negative signs. Practice basic algebra carefully.

Practice Questions

Try It Yourself

QUESTION: Find the slope of the altitude from vertex P to side QR, if the slope of QR is 3/4. | ANSWER: -4/3

QUESTION: A triangle has vertices D(2, 3), E(6, 1), and F(4, 5). Find the equation of the altitude from vertex D to side EF. | ANSWER: 2x - 2y + 2 = 0 (or x - y + 1 = 0)

QUESTION: For a triangle with vertices K(0, 0), L(4, 0), and M(2, 6), find the equation of the altitude from vertex M to side KL. | ANSWER: x - 2 = 0

MCQ

Quick Quiz

What is the slope of an altitude if the side it is perpendicular to has a slope of -5?

-1/5

2026-01-05T00:00:00.000Z

-5

The Correct Answer Is:

The slope of a line perpendicular to another line is the negative reciprocal of the original line's slope. The negative reciprocal of -5 is 1/5.

Real World Connection

In the Real World

When a drone delivers a package, its navigation system calculates the shortest, most stable path to drop the package. This often involves determining a 'perpendicular' path to the ground or a target, much like finding an altitude. Similarly, in cricket, analysts use geometry to study the angle of a ball's bounce or a fielder's throw, which involves understanding perpendicular lines.

Key Vocabulary

Key Terms

ALTITUDE: A line segment from a vertex perpendicular to the opposite side. | PERPENDICULAR: Forming a 90-degree angle. | SLOPE: The steepness or gradient of a line (rise over run). | VERTEX: A corner point of a triangle. | NEGATIVE RECIPROCAL: Flipping a fraction and changing its sign.

What's Next

What to Learn Next

Great job understanding altitudes! Next, explore the 'Orthocentre of a Triangle', which is the point where all three altitudes of a triangle intersect. This will show you how these concepts connect and form a deeper understanding of triangle properties.