Question

Question #0583a

Answer 1

see explanation.

Explanation:

Given that `angleABC=90^@`,
`=> AC` is the diameter of the circumscribed circle of `DeltaABC`.
As `BE` is perpendicular to diameter `AC`,
`=> BD=DE`

Intersecting chord theorem : when two chords intersect each other inside a circle, the products of their segment are equal.
`=> AD*DC=BD*DE=BD^2`
`=> 16*9=BD^2, => BD=12`.

By Pythagorean Theorem:
`x^2=AD^2+BD^2=16^2+12^2=400`
`=> x=sqrt400=20`

`y^2=CD^2+BD^2=9^2+12^2=225`
`=> y=sqrt225=15`

Hence, `BD=12, x=20, y=15`

Answer 2

Sol. 2. (see explanation).

Explanation:

Solution 2.

As kindly suggested by Ratnaker Mehta, the following results are very useful in such geometrical problems.

(1) `BD^2=AD*DC`
(2) `AB^2=AD*AC`
(3) `CB^2=CD*CA`

`=> BD^2=16*9=144, => BD=sqrt144=12`
`=> AB^2=16*(16+9)=400, => AB=sqrt400=20`
`=> CB^2=9*(16+9)=225, => CB=sqrt225=15`

proof :

`DeltaABC, DeltaADB and DeltaBDC` are similar triangles.

(1) `(AD)/(DB)=(BD)/(DC), => BD^2=AD*DC`

(2) `(AB)/(AC)=(AD)/(AB), => AB^2=AD*AC`

(3) `(BC)/(CA)=(DC)/(CB), => BC^2=CD*CA`