Right Triangle Altitude Theorem: Definition of Right Triangle Altitude Theorem (For CBSE, ICSE, IAS, NET, NRA 2022)

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Right Triangle Altitude Theorem

The relationship between altitude drawn on the hypotenuse from vertex of the right angle including the segments within which the hypotenuse gets divided by altitude.

Right Angled Triangle PRS

In the above figure is right angled at S

If two triangles are like each other then, we have

(i) The corresponding angles of both the triangles are equal.

(ii) The corresponding sides of both the triangles are in proportion to one another.

Hence, two triangles are similar if

(i) , and

(ii) =

Using construction in

Right Angled Triangle

In the above figure, taking

Also, as is right angled at S,

Therefore,

In by angle sum property we have,

(Using equation 1)

Similarly, in

(Angle Sum Property)

(Using equation 1)

Thus, by AA axiom of similarity,

Thus, in ΔPQS and ΔRQS,

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

Applications

  • This method allows for the construction of square roots.
  • Also, this theorem can be used to provide a geometrical proof of the AM – GM inequality in the case of two numbers.
    • Inequality of arithmetic and geometric means states that the arithmetic mean of a list of non-negative real numbers is greater than or equal to the geometric mean of the same list.
  • AM – GM inequality also includes weights or generalized means.