Question

Solve the following problems:

b

Given: ∆MOP

P∆MOP =12+4 3

m∠P = 90°, m∠M = 60°

Find: MO

Answer 1

The measure of side MO is 8 unit

Explanation:

Given as :

In the Triangle ΔMOP ,

∠P = 90°

∠M = 60°

The perimeter of ΔMOP = 12 + 4√3

Now, From figure , in Triangle ΔMOP

∠O = 180° - ( ∠P + ∠M )

or, ∠O = 180° - ( 90° + 60° )

or, ∠O = 180° - 150°

∴ ∠O = 30°

Now, from Triangle

Sin 90° =
`(\textrm perpendicular)/(\textrm hypotenuse)`

Or, Sin 90° =
`(\textrm OP)/(\textrm OM)`

Or,
`(\textrm OP)/(\textrm OM)` = 1

Again

Sin 60° =
`(\textrm perpendicular)/(\textrm hypotenuse)`

Or, Sin 60° =
`(\textrm OP)/(\textrm OM)`

Or,
`(\textrm OP)/(\textrm OM)` =
`(√(3) )/(2)`

Similarly

Sin 30° =
`(\textrm base)/(\textrm hypotenuse)`

Or, Sin 30° =
`(\textrm PM)/(\textrm OM)`

Or,
`(\textrm PM)/(\textrm OM)` =
`(1)/(2)`

So, The ratio of the sides as

PM : MO : OP = 1 : 2 :
`√(3)`

Let PM = x

MO = 2 x

OP = x
`√(3)`

Now, from question

The perimeter of triangle ΔMOP = 12 + 4√3

I.e, The sum of sides of triangle ΔMOP = 12 + 4√3

or, PM + MO + OP = 12 + 4√3

or, x + 2 x + x
`√(3)` = 12 + 4√3

or, 3 x + x
`√(3)` = 12 + 4√3

Or, x ( 3 +
`√(3)` ) = 4 ( 3 +
`√(3)` )

Now, equating both side we get

x = 4

So, The measure of side MO = 2 x

I.e The measure of side MO = 2 × 4 = 8 unit

Hence The measure of side MO is 8 unit Answer