Difference Between Altitude and Median of a Triangle (With Table & Diagram)
The concept of Altitude and Median of a Triangle is fundamental in geometry, crucial for exams, and extremely useful in understanding the properties and construction of triangles in mathematics and beyond.
What Is Altitude and Median of a Triangle?
An Altitude of a triangle is a line segment from a vertex that meets the opposite side at a right angle, showing the height of the triangle from that base. A Median is a line segment that joins a vertex to the midpoint of its opposite side, dividing the triangle into two equal-area parts. You’ll find this concept applied in topics like centroids, area calculation, and the special properties of triangles such as equilateral, isosceles, and scalene types.
Key Formula for Altitude and Median of a Triangle
Here’s the standard formula:
- Altitude (h) from side \( a \): \( h_a = \frac{2 \times \text{Area}}{a} \)
- Median (m) from side \( a \): \( m_a = \frac{1}{2}\sqrt{2b^2 + 2c^2 - a^2} \) where \( a, b, c \) are the triangle’s sides
Difference Between Altitude and Median
| Feature | Altitude | Median |
|---|---|---|
| Definition | Line from a vertex, perpendicular to the opposite side | Line from a vertex to midpoint of opposite side |
| Purpose | Shows height; used in area calculation | Divides triangle into two equal-area parts |
| Always Perpendicular? | Yes | No |
| Concurrency Point | Orthocenter | Centroid |
| Formula | \( h = \frac{2 \times \text{Area}}{\text{base}} \) | \( m = \frac{1}{2}\sqrt{2b^2+2c^2-a^2} \) |
| Number per Triangle | 3 | 3 |
How to Construct Altitude and Median
- To draw a median, locate the midpoint of a side using a ruler/compass, then connect it to the opposite vertex.
- To draw an altitude, use a set-square or protractor to drop a perpendicular from the selected vertex to the opposite side, or its extension.
Step-by-Step Illustration
- Given triangle ABC, with sides \( a = 7 \), \( b = 8 \), \( c = 9 \).
Find the median from vertex A.
- Formula: \( m_a = \frac{1}{2}\sqrt{2b^2 + 2c^2 - a^2} \).
Plug in values: \( m_a = \frac{1}{2}\sqrt{2 \times 8^2 + 2 \times 9^2 - 7^2} \)
- Compute: \( = \frac{1}{2}\sqrt{128 + 162 - 49} = \frac{1}{2}\sqrt{241} \approx \frac{1}{2} \times 15.52 \approx 7.76 \).
- Answer: The median from A is about 7.76 units.
Special Triangles: Properties
- Equilateral Triangle: Median and altitude from a vertex are the same line, all three are equal, and meet at a single point.
- Isosceles Triangle: The median and altitude from the vertex between equal sides coincide.
- Scalene Triangle: Altitudes and medians are all different.
- Right Triangle: One altitude is the side itself (the height), medians and altitudes differ except in special cases.
Try These Yourself
- Draw a triangle and mark all 3 medians. Where do they meet?
- For triangle sides 5, 6, 7, use the median formula to find the median from the side of length 5.
- Write 2 differences between altitude and median in your notebook.
- In an equilateral triangle, show by calculation that the altitude equals the median.
Frequent Errors and Misunderstandings
- Confusing the altitude with the median—remember, only the altitude is always perpendicular.
- Assuming medians always cut the base at 90° (they rarely do, except in equilateral triangles).
- Forgetting that altitude can fall outside the triangle in obtuse-angled triangles.
Relation to Other Concepts
The idea of altitude and median of a triangle connects closely with centroid (where medians meet), orthocenter (where altitudes meet), triangle area, and the properties of different triangle types. Understanding medians and altitudes makes geometry construction problems and Olympiad questions much easier.
Speed Trick or Vedic Shortcut
Here’s a quick shortcut: In an equilateral triangle of side \( a \), the length of the median (and altitude) is \( \frac{\sqrt{3}}{2}a \). Just multiply the side by 0.866 to get the answer—super fast for exam revision!
Example: Equilateral triangle with side 6.
Median = \( 0.866 \times 6 = 5.196 \).
Tricks like these are time-savers in quizzes and Olympiads. Vedantu’s live classes share more shortcuts for geometry topics.
Classroom Tip
Visual learners remember: “Median = Middle, Altitude = At 90°”. Drawing each with different colored pens in your triangle sketch brings instant clarity. Vedantu’s teachers often use color-coding to help students remember during lessons.
Cross-Disciplinary Usage
Altitude and median of a triangle is not only critical in Maths for geometry and construction but is also applied in Physics (center of mass, shortest path), Engineering (bridge design), and Computer Science (graphics and algorithms). JEE and NEET aspirants frequently encounter these concepts.
Wrapping It All Up
We explored altitude and median of a triangle—how to define, construct, and calculate them, their special cases for different triangles, and why they matter in higher studies and real-world logic. Vedantu offers further practice, worksheets, and solved problems to give students the confidence needed for exams and Olympiads.
Further Reading and Related Topics
FAQs on Altitude and Median of a Triangle Explained with Examples
1. What is the median of a triangle?
A median of a triangle is a line segment drawn from a vertex to the midpoint of the opposite side. Each triangle has three medians, one from each vertex. The medians intersect at a point called the centroid. Importantly, a median divides the triangle into two smaller triangles of equal area.
2. What is the altitude of a triangle?
An altitude of a triangle is a perpendicular line segment drawn from a vertex to the opposite side (or its extension). It represents the height of the triangle relative to that base. Like medians, every triangle possesses three altitudes, one from each vertex. These altitudes intersect at a point known as the orthocenter.
3. What is the difference between the altitude and median of a triangle?
The key difference lies in their construction and properties. A median connects a vertex to the midpoint of the opposite side, bisecting that side. An altitude is a perpendicular line from a vertex to the opposite side, representing the height. While a median always divides the triangle into two equal-area triangles, an altitude doesn't necessarily bisect the opposite side. In special cases like equilateral triangles, the altitude and median are identical.
4. How do I find the altitude of a triangle?
The method for calculating the altitude depends on the information available. If you know the area (A) and base (b) of the triangle, use the formula: Altitude = 2A/b. Alternatively, trigonometric functions (like sine) can be used if you know angles and side lengths.
5. How do I find the median of a triangle?
To find a median, you need to locate the midpoint of one side. You can use the midpoint formula if you have the coordinates of the endpoints. Then, draw a line segment connecting that midpoint to the opposite vertex. The length of the median can be calculated using the distance formula or trigonometric methods depending on given information.
6. Are the altitude and median always the same in a triangle?
No, the altitude and median are only the same in an isosceles triangle when the altitude is drawn to the unequal side and in an equilateral triangle. In other triangle types, they are distinct line segments.
7. What is the relationship between the medians and the centroid?
The three medians of a triangle are always concurrent, meaning they intersect at a single point. This point of intersection is called the centroid. The centroid divides each median into a 2:1 ratio.
8. What is the relationship between the altitudes and the orthocenter?
The three altitudes of a triangle always intersect at a single point called the orthocenter. The location of the orthocenter varies depending on the type of triangle (e.g., inside an acute triangle, outside an obtuse triangle, on the triangle for a right-angled triangle).
9. How are medians and altitudes used in solving problems?
Medians are useful for finding the area of a triangle, determining the centroid, and solving problems related to dividing the triangle into equal parts. Altitudes are crucial for calculating the area, finding the orthocenter, and solving problems involving the height of the triangle.
10. Can the altitude and median coincide in a triangle?
Yes, they can coincide. This happens in isosceles triangles where the altitude is drawn to the unequal side and in equilateral triangles, where all altitudes and medians are congruent and coincide.
11. How do I draw the altitude and median of a triangle?
To draw a median, find the midpoint of a side and connect it to the opposite vertex. To draw an altitude, from a vertex, construct a perpendicular line to the opposite side using a compass and straightedge (or protractor and ruler).
12. What are some real-world applications of medians and altitudes?
Medians and altitudes find applications in various fields including architecture (e.g., roof design), engineering (e.g., bridge construction), and surveying (e.g., land measurement). They are fundamental concepts in geometry and have broader implications in more advanced mathematical and scientific applications.
