Triangles • Topic 3 of 3

Medians, Altitudes, and Construction of Triangles

What is a Median? A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side. Every triangle has three medians, and they intersect at the centroid.

Properties of Medians:

A median divides the triangle into two triangles of equal area
The centroid divides each median in the ratio $2:1$ (vertex to centroid : centroid to midpoint)

What is an Altitude? An altitude of a triangle is a perpendicular line segment from a vertex to the opposite side (or its extension). Every triangle has three altitudes, and they intersect at the orthocenter.

Medians vs Altitudes:

Feature	Median	Altitude
Goes from	Vertex to opposite side	Vertex to opposite side
Special property	Meets at midpoint	Perpendicular to opposite side
Intersection point	Centroid	Orthocenter

Construction of Triangles (Using Ruler and Compass):

Given conditions	Construction steps
SSS (three sides)	Draw base, draw arcs from ends with other side lengths
SAS (two sides and included angle)	Draw base, construct angle, mark second side
ASA (two angles and included side)	Draw base, construct angles at ends, extend to meet
RHS (hypotenuse and one side)	Draw side, construct perpendicular, draw hypotenuse arc

Solved Examples

Worked Example

In $\triangle ABC$, $D$ is the midpoint of $BC$. If $AD$ is a median and $AB = AC$, what type of triangle is it?

Solution- $AB = AC$ means the triangle is isosceles - Median to the base of an isosceles triangle is also the altitude - So $AD$ is perpendicular to $BC$ - **Answer:** Isosceles triangle with $AB = AC$ *Example 2: Construct a triangle with sides 4 cm, 5 cm, and 6 cm. Solution: - Draw base $BC = 6$ cm - From $B$, draw arc of radius 5 cm - From $C$, draw arc of radius 4 cm - Mark intersection as $A$ - Join $AB$ and $AC$ - **Answer:** Triangle $ABC$ constructed *Example 3: In a triangle, the centroid divides a median in the ratio $2:1$. If the length of the median is 12 cm, find the distance from the vertex to the centroid. Solution: - Let the median be $AD$ with centroid $G$ - $AG : GD = 2 : 1$ - Total parts = $2 + 1 = 3$ - $AG = \frac{2}{3} \times 12 = 8$ cm - $GD = \frac{1}{3} \times 12 = 4$ cm - **Answer:** 8 cm from vertex to centroid

Key Points

Median: vertex to midpoint of opposite side; three medians meet at centroid
Altitude: vertex perpendicular to opposite side; three altitudes meet at orthocenter
In an isosceles triangle, the median to the base is also the altitude
Centroid divides each median in ratio $2:1$ (vertex : midpoint)
Triangles can be constructed using SSS, SAS, ASA, or RHS conditions
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Quick Check — 4 MCQs

Tap an option to check your answer0 / 4

Q1.A median joins a vertex to the ___ of the opposite side.

Explanation: Midpoint.

Q2.An altitude is ___ to the opposite side.

Explanation: Perpendicular.

Q3.The three medians meet at the:

Explanation: Centroid.

Q4.The centroid divides each median in the ratio:

Explanation: $2:1$.

Practice more MCQs → 📝 Assignment · 35 marks