Question

In triangle ABC, AD is perpendicular to BC. sin B = 0.8, BD = 9 cm and tan C = 1.
Find the length of AB, AD, AC, and DC.

Answer 1

Consider the figure below :

sin B = `(8)/(10) = (4)/(5)`

i.e.`"perpendicular"/"hypotenuse" = "AD"/"AB" = (4)/(5)`

Therefore if length of perpendicular = 4x, length of hypotenuse = 5x

Since
AD² + BD² = AB²  ...[ Using Pythagoras Theorem ]

(5x)² – (4x)² = BD²

BD² = 9x²

∴ BD = 3x 

Now
BD = 9

3x = 9

x = 3

Therefore
AB = 5x

= 5 x 3

= 15 cm

And
AD= 4x

= 4 x 3

= 12 cm

Again
tan C = `(1)/(1)`

i.e.`"perpendicular"/"base" = "AD"/"DC" = (1)/(1)`

Therefore if length of perpendicular = x, length of base = x

Since
AD² + DC² = AC² ...[ Using Pythagoras Theorem ]

(x)² + (x)² = AC²

AC² = 2x² 

∴ AC = `sqrt2x`

Now
AD = 12

x = 12

Therefore
DC = x

= 12 cm

And
AC = `sqrt2`

= `sqrt2` x 12

= 12`sqrt2"cm"`