Medians and Altitudes of Triangle
Every triangle has special line segments drawn from its vertices. Two of the most important are medians and altitudes.
A median connects a vertex to the midpoint of the opposite side. An altitude is the perpendicular from a vertex to the opposite side (or its extension).
Every triangle has 3 medians and 3 altitudes. The three medians meet at a single point called the centroid. The three altitudes meet at a point called the orthocentre.
What is Medians and Altitudes of Triangle - Grade 7 Maths (Triangles)?
- A line segment joining a vertex to the midpoint of the opposite side.
- Every triangle has exactly 3 medians.
- The three medians always intersect at a single point called the centroid (G).
- The centroid divides each median in the ratio 2:1 from the vertex.
Altitude:
- The perpendicular distance from a vertex to the opposite side (or its extension).
- Every triangle has exactly 3 altitudes.
- The three altitudes always intersect at a single point called the orthocentre (H).
- An altitude may lie inside (acute triangle), on a side (right triangle), or outside (obtuse triangle).
Medians and Altitudes of Triangle Formula
Centroid divides each median in ratio 2:1:
AG:GD = 2:1
where G is the centroid, A is the vertex, and D is the midpoint of BC.
Area using altitude:
Area = (1/2) × base × altitude
Types and Properties
Position of Altitudes by Triangle Type:
- Acute triangle: All 3 altitudes lie inside the triangle. Orthocentre is inside.
- Right triangle: Two altitudes are the legs of the triangle. Orthocentre is at the right-angle vertex.
- Obtuse triangle: Two altitudes fall outside the triangle (on extensions of sides). Orthocentre is outside.
Medians always lie inside the triangle, regardless of the type of triangle.
Solved Examples
Example 1: Finding the Centroid Ratio
Problem: In triangle ABC, median AD has length 9 cm. Find AG and GD where G is the centroid.
Solution:
- Centroid divides median in ratio 2:1 from vertex.
- AG = (2/3) × 9 = 6 cm
- GD = (1/3) × 9 = 3 cm
Answer: AG = 6 cm, GD = 3 cm.
Example 2: Finding Area Using Altitude
Problem: A triangle has base 12 cm and altitude to that base is 8 cm. Find the area.
Solution:
- Area = (1/2) × 12 × 8 = 48 cm²
Answer: Area = 48 cm².
Example 3: Identifying Altitudes in Right Triangle
Problem: In right triangle PQR with right angle at Q, identify the three altitudes.
Solution:
- From Q: the altitude to PR (the hypotenuse) — lies inside.
- From P: the altitude is QP itself (perpendicular to QR since ∠Q = 90°). Actually, PQ is the leg perpendicular to QR.
- From R: the altitude is QR itself (perpendicular to PQ).
Answer: Two altitudes are the legs PQ and QR. The third is the perpendicular from Q to the hypotenuse PR.
Example 4: Median Length Calculation
Problem: In a triangle, a median from vertex A to side BC. BC = 10 cm, so D (midpoint of BC) gives BD = DC = 5 cm. If the centroid G is 4 cm from A along the median, find the full length of the median.
Solution:
- AG = 4 cm. Centroid divides in 2:1.
- AG:GD = 2:1, so GD = 4/2 = 2 cm.
- Full median AD = AG + GD = 4 + 2 = 6 cm.
Answer: The median is 6 cm long.
Example 5: Finding Altitude
Problem: Area of a triangle is 60 cm² and one side is 15 cm. Find the altitude to that side.
Solution:
- Area = (1/2) × base × height
- 60 = (1/2) × 15 × h
- 60 = 7.5h
- h = 8 cm
Answer: The altitude is 8 cm.
Real-World Applications
Real-world uses:
- Centre of gravity: The centroid is the balance point of a triangular shape. Engineers use it to find where to support a triangular plate.
- Architecture: Altitudes determine the height of triangular roofs and gables.
- Navigation: Finding the centroid of a triangular region on a map.
Key Points to Remember
- A median joins a vertex to the midpoint of the opposite side.
- An altitude is the perpendicular from a vertex to the opposite side.
- Every triangle has 3 medians and 3 altitudes.
- Medians meet at the centroid (divides each median 2:1 from vertex).
- Altitudes meet at the orthocentre.
- Altitudes may lie outside the triangle (obtuse case).
- Medians always lie inside the triangle.
- Area = (1/2) × base × corresponding altitude.
Practice Problems
- A median is 12 cm long. Find the distances from the centroid to the vertex and to the midpoint.
- Find the area of a triangle with base 14 cm and altitude 9 cm.
- Area = 75 cm², base = 25 cm. Find the altitude.
- Where is the orthocentre in a right triangle?
- In an equilateral triangle of side 6 cm, find the length of a median.
Frequently Asked Questions
Q1. What is a median of a triangle?
A line segment from a vertex to the midpoint of the opposite side. Every triangle has 3 medians.
Q2. What is the centroid?
The point where all 3 medians intersect. It divides each median in the ratio 2:1 from the vertex. It is the centre of gravity of the triangle.
Q3. What is an altitude of a triangle?
The perpendicular from a vertex to the opposite side (or its extension). It represents the height of the triangle from that vertex.
Q4. Can an altitude fall outside the triangle?
Yes. In an obtuse triangle, the altitudes from the two acute vertices fall outside the triangle, on extensions of the opposite sides.
Q5. Are median and altitude the same?
Not in general. A median goes to the midpoint; an altitude is perpendicular to the opposite side. They coincide only in special cases like an equilateral triangle or the median/altitude from the vertex angle of an isosceles triangle.
Related Topics
- Angle Sum Property of Triangle
- Classification of Triangles
- Area of Triangle
- Perpendicular Lines
- Exterior Angle Property of Triangle
- Properties of Isosceles Triangle
- Properties of Equilateral Triangle
- Triangle Inequality Property
- Right-Angled Triangle Property
- Congruent Triangles - Proofs
- Inequalities in Triangles
- Similar Triangles
- Basic Proportionality Theorem (BPT)
- Converse of Basic Proportionality Theorem
