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:
[tex]m\angle MON=m\angle NOP+m\angle POM[/tex]
As ∠NOP = ∠POM, we get:
[tex]m\angle MON=m\angle NOP+m\angle NOP=2\cdot m\angle NOP[/tex]
Replacing the value we get:
[tex]m\angle MON=2\cdot20=40[/tex]
Based on this, we can use the trigonometric functions, as we have an angle and one side. Specifically, the tangent function:
[tex]\tan \alpha=\frac{opposite}{\text{adyacent}}[/tex]
First, to calculate NP, we get the following:
[tex]\tan 20=\frac{NP}{6}[/tex]
Isolating for NP:
[tex]NP=6\cdot\tan 20[/tex][tex]NP=2.18[/tex]
Then, calculating for MN we get the following:
[tex]\tan 40=\frac{MN}{6}[/tex]
Isolating for MN:
[tex]MN=6\cdot\tan 40[/tex][tex]MN=5.03[/tex]
Answer:
• NP = 2.18
,
• MN = 5.03
