Question

In the triangle ABC, BD bisects angle B and is perpendicular to AC. If the lengths of the sides of the triangle are expressed in terms of x and y as shown, find the values of x and y.

Answer 1

In ΔABD aand ΔDBC,
BD  = BD                 ...[ Common ]
∠BDA = ∠BDC        ...[ each equal to 90° ]
∠ABD = ∠DBC        ...[ BD bisects ∠ABC ]
∴ ΔABD ≅ ΔDBC       ...[ ASA criterion ]

Therefore,
AD = DC
x + 1 = y + 2
⇒ x = y + 1               ..... (i)
and AB = BC
3x + 1 = 5y - 2

Subtituting the value of x from (i)
3( y + 1 ) + 1 = 5y - 2 
⇒ 3y + 3 + 1 = 5y - 2
⇒ 3y +  4 = 5y - 2
⇒ 2y = 6
⇒ y = 3

Putting y = 3 in (i)
x = 3 + 1 
∴ x = 4