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galben [10]
3 years ago
7

Solve the following problems:

Mathematics
1 answer:
lorasvet [3.4K]3 years ago
5 0

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°

<u>Now, from Triangle</u>

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

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