In this module, we will learn about the "Altitude and Median of a Triangle�.

What is a Triangle?

A triangle is a simple closed curve made of three line segments.

It has three vertices, three sides, and three angles.

Here is a triangle PQR.

It has three sides PQ, QR, and RP.

Three angles are angle QPR, angle PQR and angle QRP. And it has three vertices P, Q and R.

Sides: PQ, QR , RP
Angles: ?QPR, ?PQR, ?QRP
Vertices: P, Q, R

Now, let us explore the altitude and median of a triangle.

First, we shall discuss �Medians of a triangle�.

A Median connects a vertex of a triangle to the mid-point of the opposite side.

The line segment AD, joining the mid-point of BC to its opposite vertex A is called a Median of the triangle.

Let us discuss something more about the median of a triangle.

• How many medians can a triangle have?

A triangle can have only three medians. In triangle ABC, we can only draw three medians from three vertices A, B and C respectively.

AD, BE and CF are medians of triangle ABC from vertices A, B, and C respectively.

• Does a median lie wholly in the interior of the triangle?

Yes, a median lies wholly in the interior of the triangle.

Now let us discuss �Altitudes of a Triangle� in this blog on "Altitude and Median of a Triangle."

An Altitude has one endpoint at a vertex of the triangle and the other on the line containing the opposite side. Also, the altitude is perpendicular to the opposite side.

Through each vertex, an altitude can be drawn. In DPQR, PM is the altitude of triangle PQR from vertex P to opposite side QR.

Let us discuss more the Altitudes of a triangle.

How many altitudes can a triangle have?

A triangle can have only three altitudes. In triangle ABC, we can only draw three altitudes from three vertices A, B, and C respectively.

AD, BE and CF are altitudes of triangle ABC from vertices A, B, and C respectively.

Will an altitude always lie in the interior of a triangle? If you think that this need not be true, draw a rough sketch to show such a case.

No, altitude will not always lie in the interior of the triangle.

We can consider the following cases

• In the acute-angled triangle ABC, AD is the altitude and is in the interior of the triangle ABC.
• In the right-angled triangle ABC, AC, and BC are altitudes of right-angled triangle ABC.
• But in obtuse-angled triangle ABC, AD is the median of triangle ABC and it is not in the interior of the triangle. That is why we say that altitude always does not always lie inside the triangle.

• Can you think of a triangle in which two altitudes of the triangle are two of its sides?

Yes, in the right-angled triangle ABC, there are two altitudes of triangle AC and BC.

• Can the altitude and median be the same for a triangle?

Probably answer is yes, In a equilateral triangle altitudes and the medians are the same.

Read More:

Exterior Angle Property of a Triangle: Proof Of Exterior Angle Theorem

Angle Sum Property of a Triangle: Theorem - Definition With Proof