Question

I'm having trouble finding the length of NP and MN, thinking it has something to do with tan, cos, and sin, but not completely sure.

Answer 1

Bisects: to divide into two equal parts.

In this case, DB is bisecting the ∠ABC, then the ∠ABD

As OP is bisecting ∠MON, that means that ∠NOP and ∠POM have the same measure.

Then:

`m\angle MON=m\angle NOP+m\angle POM`

As ∠NOP = ∠POM, we get:

`m\angle MON=m\angle NOP+m\angle NOP=2\cdot m\angle NOP`

Replacing the value we get:

`m\angle MON=2\cdot20=40`

Based on this, we can use the trigonometric functions, as we have an angle and one side. Specifically, the tangent function:

`\tan \alpha=\frac{opposite}{\text{adyacent}}`

First, to calculate NP, we get the following:

`\tan 20=(NP)/(6)`

Isolating for NP:

`NP=6\cdot\tan 20`
`NP=2.18`

Then, calculating for MN we get the following:

`\tan 40=(MN)/(6)`

Isolating for MN:

`MN=6\cdot\tan 40`
`MN=5.03`

Answer:

• NP = 2.18

,

• MN = 5.03