Question

Solve the following problems:

Answer 1

Answer:

The measure of side MO is 8 unit

Step-by-step 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° = $\dfrac{\textrm perpendicular}{\textrm hypotenuse}$

Or, Sin 90° = $\dfrac{\textrm OP}{\textrm OM}$

Or,  $\dfrac{\textrm OP}{\textrm OM}$ = 1

Again

Sin 60° = $\dfrac{\textrm perpendicular}{\textrm hypotenuse}$

Or, Sin 60° = $\dfrac{\textrm OP}{\textrm OM}$

Or,  $\dfrac{\textrm OP}{\textrm OM}$ = $\dfrac{\sqrt{3} }{2}$

Similarly

Sin 30° = $\dfrac{\textrm base}{\textrm hypotenuse}$

Or, Sin 30° = $\dfrac{\textrm PM}{\textrm OM}$

Or,  $\dfrac{\textrm PM}{\textrm OM}$ = $\frac{1}{2}$

So, The ratio of the sides as

PM : MO : OP = 1 : 2 : $\sqrt{3}$

Let  PM = x

MO = 2 x

OP = x$\sqrt{3}$

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$\sqrt{3}$ = 12 + 4√3

or, 3 x  +  x$\sqrt{3}$ = 12 + 4√3

Or, x ( 3 +$\sqrt{3}$ ) = 4 ( 3 +$\sqrt{3}$ )

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