Geometrical Property of Altitudes

Here we will discuss about a geometrical property of altitudes.

The three altitudes of triangle are concurrent. The point at which they intersect is known as the orthocentre of the triangle.

In the adjoining figure, the three altitudes XP, YQ and ZR intersect at the orthocentre O.

In the adjoining figure ∆XYZ is a right-angled triangle. Here we see that XY ⊥ YZ, YZ ⊥ XY and YM ⊥ XZ.

Thus, the three altitudes XY, YZ and YM intersect at Y. So, Y is the orthocentre of ∆XYZ.

In an obtuse-angled ∆XYZ, the altitudes with respect to XY, YZ and ZX are ZR, XP and YQ respectively.

The altitudes when produced, meet at O. The O is the orthocentre of ∆XYZ.

We see that the orthocentre may lie within, on or outside the triangle.

9th Grade Math

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