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.

#### Solution

Δ PQR is a right triangle, right angled at Q

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

`6+z^2=16+x^2+8x`

`z^2-x^2-8x=16-36`

`z^2-x^2-8x=16-36`

`z^2-x^2-8x=-20`......(1)

Δ QSP is a right triangle right angled at S

`QS^2+PS^2=PQ^2`

`y^2+4^2=6^2`

`y^2+16=36`

`y^2=36-16`

`y^2=20`

`y=sqrt20`

`y=sqrt(2xx2xx5)`

`y=2sqrt5`

Δ QSR is a right triangle right angled at S

`QS^2+RS^2=QR^2`

`y^2+x^2=z^2`..........(2)

Now substituting `y^2+x^2=z^2` in equation (i) we get

`y^2+x^2-x^2-8x=-20`

`20-8x=-20`

`-8x=-20-20`

`-8x=-40`

`x=40/8`

`x=5`

Now substituting ` x = 5` and `y^2=20` in equation (ii) we get

`y^2+x^2=z^2`

`20+5^2=z^2`

`20+25=z^2`

`45=z^2`

`sqrt(3xx3xx5)=z^2`

`3sqrt5=z`

Hence the value of x, y and z are `5,2sqrt5,3sqrt5`