Question

In a right triangle, the angle bisector of an acute angle divides the opposite side into
segments of length 1 and 2. What is the length of the bisector?
my teatcher said it should be √4 or √5 but how can you tell the way

Answer 1

let ABC be right angled triangle with angle B = 90 degrees and AD be the angle bisector.

therefore, BD = 1cm (according to the question)

DC = 2cm

BC = 1cm + 2cm = 3cm

AB:AC = 1:2 (An angle bisector divides
the opposite side in the
ratio of the other two sides)
let AB be x cm and AC be 2x cm

(AC)^2 = (AB)^2 + (BC)^2

=> (2x)^2 = x^2 + (3cm)^2
=> 4x^2 = x^2 + 9
=> 4x^2-x^2 = 9
=> 3x^2 = 9
=> x^2 = 3
=> x = root 3

AB = x = root 3 cm
BD = 1 cm
(AD)^2 = (AB)^2 + (BD)^2

=> (AD)^2 = (root 3)^2 + 1^2
=> (AD)^2 = 3+1
=> (AD)^2 = 4
=> AD = root 4 cm
=> AD = 2cm (Ans)