Question

In each of the figures given below, an altitude is drawn to the hypotenuse by a right-angled triangle. The length of different line-segment are marked in each figure. Determine x, y, z in each case.

Answer 1

 Δ ABC is right angled triangle right angled at B

`AB^2+BC^2= AC ^2`

`x^2+z^2=(4+5)^2`

`x^2+z^2=9^2`

`x^2+z^2=81`...............(1)

 Δ BAD  is right triangle right angled at D

`BD^2+AD^2=AB^2`

`y^2+4^2=x^2`

`y^2+16=x^2`

`16=x^2-y^2`...............(2)

 Δ BDC  is right triangle right angled at D

`BD^2+DC^2=BC^2`

`y^2+25=z^2`

`25 = z^2-y^2`..................(3)

By canceling equation (1) and (3) by elimination method, we get

y canceling and by elimination method we get

`z^2=90/2`

`z^2=45`

`z=sqrt45`

`z=sqrt(3xx3xx5)`

`z=3sqrt5`

Now, substituting  `z^2=45` in equation (iv) we get

`y^2+z^2=65`

`y^2+45=65`

`y^2=65-45`

`y^2=20`

`y = sqrt20`

`y=sqrt(2xx2xx5)`

`y=2sqrt5`

Now, substituting  `y^2=20` in equation (ii) we get

`x^2-y^2=16`

`x^2-20=16`

`x^2=16+20`

`x^2=36`

`x=sqrt36`

`x=sqrt(6xx6)`

`x = 6`

Hence the values of x, y, z is  `6,2sqrt5,3sqrt5`