Altitudes of a triangle :

The perpendicular segment drawn from a vertex of a triangle on the side opposite to it is called an altitude of the triangle.

In Δ ABC, seg AP is an altitude on the base BC.

Construction of altitudes of a triangle :

(1) Draw any Δ XYZ.

(2) Draw a perpendicular from vertex X on the side YZ using a set - square. Name the point where it meets side YZ as R. Seg XR is an altitude on side YZ.

(3) Considering side XZ as a base, draw an altitude YQ on side XZ. seg YQ ⊥ seg XZ.

(4) Consider side XY as a base, draw an altitude ZP on seg XY. seg ZP ⊥ seg XY. seg XR, seg YQ, seg ZP are the altitudes of Δ XYZ.

Note that, the three altitudes are concurrent.

The point of concurrence is called the orthocenter of the triangle. It is denoted by the letter ‘O’.

Remember :

(i) A triangle has three altitudes.

(ii) The altitudes of a triangle pass through exactly one point, that means they are concurrent.

(iii) The point of concurrence of the three altitudes of the triangle is called the orthocentre and is usually denoted by O.

(iv) The orthocentre of a right angled triangle is the vertex containing the right angle.

In a right-angled triangle:

• One altitude is AB (perpendicular to BC)
• Second altitude is BC (perpendicular to AB)
• The third altitude is drawn from B to the hypotenuse AC

So, all three altitudes meet at vertex B.

(v) The orthocentre of an obtuse angled triangle is in the exterior of the triangle.

• In an obtuse-angled triangle, the altitudes do not intersect inside the triangle.
• However, the lines containing the altitudes are concurrent and meet at a point outside the triangle.

(vi) The orthocentre of an acute angled triangle is in the interior of the triangle.

Median of a triangle :

The segment joining the vertex and midpoint of the opposite side is called the median of a triangle.

In Δ ABC, M is the midpoint of side BC. ∴ seg AM is the median.

Construction of medians of a triangle :

(1) Draw Δ PQR.

(2) Find the midpoint A of side QR by constructing the perpendicular bisector of seg QR. Draw seg PA. Seg PA is the median.

(3) Find the midpoint B of side PQ by constructing the perpendicular bisector of seg PQ. Draw seg RB. Seg RB is the median.

(4) Find the midpoint C of side PR by constructing the perpendicular bisector of seg PR. Draw seg QC. Seg QC is the median.

(5) Seg PA, seg RB and seg QC are the medians of Δ PQR.

(6) Note that the three medians are concurrent.

Mark the point of concurrence as G.

Remember :

• A triangle has three medians.
• The medians of a triangle are concurrent.
• Their point of concurrence is called the centroid and it is usually denoted by G.
• For any triangle, the location of centroid G is always in the interior of the triangle.
• The centroid divides each median in the ratio 2 : 1.