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`
