Right Triangle Altitude Theorem:

This theorem describes the relationship between altitude drawn on the hypotenuse from vertex of the right angle and the segments into which hypotenuse is divided by altitude.

Consider a right angled triangle, \(∆ABC\)

If two triangles are similar to each other then,

i) Corresponding angles of both the triangles are equal and

ii) Corresponding sides of both the triangles are in proportion to each other.

Thus, two triangles \(ΔABC\)

i) \(∠A\)

ii) \(\frac{AB}{PQ}\)

Using constructions in \(ΔABC\)

From fig. 2, in \(ΔADC\)

\(∠ADC\)

As \(ΔABC\)

\(∠C\)

In \(∆ADC\)

\(∠ADC + ∠DCA + ∠CAD\)

\(\Rightarrow\)

Similarly in \(∆BDC\)

\(∠BDC + ∠DCB + ∠CBD\)

\(\Rightarrow∠CBD\)

Thus by AA axiom of similarity

\(∆ADC~∆CDB\)

Thus in \(ΔADC\)

\(\frac{CD}{BD}\)

\(\Rightarrow~\frac{h}{y}\)

\(\Rightarrow~h^2\)

Thus, in a right angle triangle the altitude on hypotenuse is equal to the geometric mean of line segments formed by altitude on hypotenuse.

The converse of above theorem is also true which states that any triangle is a right angled triangle, if altitude is equal to the geometric mean of line segments formed by the altitude.

The above theorem can be easily comprehended by visualizing it.

This figure can be represented by splitting \(ΔADC \)

To complete the triangles in fig 4.1 and 4.2, a square of area \(h^2\)

\(\Rightarrow~h^2\)

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