Question

In circle S,diameter PN is drawn .If measurement arc MN=110 degrees and PS=10 then find the length of MN to the nearest tenth.Show how you found your answer

Answer 1

`MN=16.4\ units`

Explanation:

step 1

Find PN

we know that

PN is a diameter

PS is a radius

`PS=10\ units`

so

`PN=2*PS=2*10=20\ units`

step 2

Find the measure of arc MP

`arc\ MN+arc\ MP=180\°`

`arc\ MN=110\°`

`arc\ MP=180\°-110\°=70\°`

step 3

Find the measure of angle MNP

we know that

The inscribed angle measures half that of the arc comprising

`<MNP=(1)/(2)(arc\ MP)`

substitute the value

`<MNP=(1)/(2)(70\°)=35\°`

step 4

Find the measure of angle MPN

we know that

The inscribed angle measures half that of the arc comprising

`<MPN=(1)/(2)(arc\ MN)`

substitute the value

`<MNP=(1)/(2)(110\°)=55\°`

step 5

Find the measure of angle NMP

The sum of the internal angles of a triangle must be equal to 180 degrees

`<NMP+55\°+35\°=180\°`

`<NMP=90\°`

therefore

The triangle MNP is a right triangle

step 6

Find the length MN

In the right triangle MNP

`sin(<MPN)=(MN)/(PN)`

substitute the values

`sin(55\°)=(MN)/(20)`

`MN=(20)sin(55\°)=16.4\ units`