Altitude of an Equilateral Triangle is also a Median – Prove | Altitude, Median – Definition, Properties

by

A triangle is a three-sided polygon that has 3 angles, 3 sides and 3 vertices. Depending on the length of sides, triangles are classified into three types. They are equilateral triangle, isosceles triangle, and scalene triangle. Have a look at the definition and properties of triangle altitude, median. Prove for the altitude of an equilateral triangle is a median is mentioned here.

An Altitude of an Equilateral Triangle is also a Median

Altitude is the line segment from the vertex that is perpendicular to the opposite side and median is the line from the vertex that divides the opposite side into two equal parts. In an equilateral triangle, altitude and median are the same. Here we will prove that an altitude of an equilateral triangle is also a median.

In the below figure, to prove AL is a median, take AB = AC and AL ⊥ BC
By drawing AL, two triangles are formed △ABL, △ALC
AB = AC [It is given]
AL = AL [Common side]
∠ALB = ∠ALC = 90° [AL ⊥ BC]
∆ALB ≅ ∆ALC by RHS criterion
So, BL = LC
AL is a median
As AL bisects BC, AL is an altitude.
Hence, proved
It is proved that an altitude of an equilateral triangle is also a median.

Altitude of an Equilateral Triangle is also a Median 3

Do Check:

Median of a Triangle – Definition, Properties

A median of a triangle is defined as a line segment that connects a vertex to the midpoint of the side that is opposite to that vertex. In the below figure, the line drawn from vertex A to BC is called altitude. Here, altitude AL divides BC into two parts, BL = LC. The important properties of a median are given here:

  • Every triangle has exactly 3 medians each from one vertex.
  • Each median of a triangle divides the triangle into two smaller triangles that have equal area.
  • Thus, 3 medians divide a triangle into 6 smaller triangles.
  • The three medians of a triangle always meet at a point.
  • The point of intersection of medians is called the centroid of the triangle.

Altitude of an Equilateral Triangle is also a Median 2

What is the Altitude of Triangle?

An altitude is defined as the perpendicular line that is drawn from the vertex to the opposite side of the triangle. In △ABC, a perpendicular line segment drawn from vertex B to opposite side AC is called altitude. Like median, triangle altitude also has some properties.

  • Every triangle can have at least 3 altitudes, one from each vertex.
  • All three altitudes of a triangle intersect at a point irrespective of triangle shape. That point is called ortho-centre of the triangle.
  • Altitude is the shortest distance from the vertex to its opposite side.

Altitude of an Equilateral Triangle is also a Median 1

FAQ’s on Altitude, Median of an Equilateral Triangle are Same

1. What is the difference between the altitude and median of a triangle?

  • The Median is the line segment drawn from vertex to the middle of the opposite side of the triangle. Whereas altitude is the perpendicular line drawn from vertex to the triangle opposite side.
  • The intersection of 3 medians is the centroid of the triangle. The intersection of three altitudes is the orthocenter of the triangle.
  • Median divides the triangle into two equal parts. But altitude does not divide the triangle into two equal parts.

2. Can the median of a triangle also be an altitude?

For an equilateral triangle, median and altitude are the same.

3. What is the altitude of an equilateral triangle, if its side length is 5 cm?

Altitude of an equilateral triangle h = s√3/2
= 5√3/2
= 2.5(√3) = 4.33

Leave a Comment