Question

Geometry: Similarities?

Answer 1

`RO=10`

Explanation:

I assume that `angle1=angle2`
`therefore SQ||OP" "` (`angle1," "angle2" are corresponding angles"`)

Using Basic Proportionality Theorem or say BPT
`=>4/x=x/9`

`=>4*9=x*x`

`=>36=x^2`

`sqrt36=x`

`6=x" "` (x is length so, can't be negative)

`RO=4+x=4+6=10`

`color(purple)(RO=10`

Answer 2

`RO=10` units

Explanation:

given `angleRSQ=angleROP, => DeltaRSQ and DeltaROP` are similar,
let `SO=RQ=x`,
`=> (RO)/(RS)=(RP)/(RQ)`
`=> (4+x)/4=(x+9)/(x)`
`=> 4x+x^2=4x+36`
`=> x^2=36`
`=> x=+-6`
as `x` can not be negative,
`=> x=6` units
`=> RO=4+x=4+6=10` units