Chapter 8.2, Problem 7MRE

To determine

To fill the blanks: with the appropriate name of the segments with reference to the given figure

Answer to Problem 7MRE

  $AX=$

  $BX$ = $CX$

Explanation of Solution

Given Information: A triangle with a right angle and a perpendicular directed to the hypotenuse which bisects it.

For a right triangle ABC with a perpendicular CX being drawn to the hypotenuse AB (as shown below), the relation between the altitude and the two segments of the hypotenuse is expressed as,

  

  $C{X}^2=AX\times BX$ ……… (1)

But, it is given that; the perpendicular drawn to the hypotenuse bisects it. Thus,

  $AX=BX$

Considering this relation in (1), it can be rewritten as,

  $C{X}^2=AX\times AX$

  $CX=AX$ ……….. (2)

It can also be written as,

  $C{X}^2=BX\times BX$

  $CX=BX$ ……….. (3)

From (2) and (3), it can be concluded that,

  $AX=BX=CX$