Question

What's the length of the |BC| ?

Answer 1

`BC=3sqrt10` units

Explanation:

Consider `DeltaABH`, by Pythagorean Theorem,
`AB^2=AH^2+BH^2`
`=> AB=sqrt(3^2+6^2)=sqrt45=3sqrt5`
As `DeltaABH and DeltaDBA` are similar,
`=> (AD)/(AB)=(HA)/(HB)`
`=> AD=(HA)/(HB)*AB=3/6*3sqrt5=(3sqrt5)/2`
Given `AD=DC, => AC=2AD=3sqrt5`
As `AC=AB, and angleA=90^@`,
`=> angleABC=angleACB=45^@`
`=> DeltaABC` is an isosceles right triangle.
`=> BC=ABsqrt2=3sqrt5*sqrt2=3sqrt10`