We have a diagram for chords in a circle.
Given that OE = 25, AB = 40 and CD = 30, we have to find the length of the other segments in the list.
We start with MB, which will be half the length of AB because OE is a perpendicular bisector.
Then:
[tex]MB=\frac{AB}{2}=\frac{40}{2}=20[/tex]
For OB we can consider that it is a radius, as O is the center and B is a point on the circumference. As the radius is constant in a circle and we know that OE = 25, then OB should also be OB = 25.
Given OB = 25 ad MB = 20, we can use the Pythagorean theorem to find OM as:
[tex]\begin{gathered} OM^2+MB^2=OB^2 \\ OM^2=OB^2-MB^2 \\ OM^2=25^2-20^2 \\ OM^2=625-400 \\ OM^2=225 \\ OM=\sqrt{225} \\ OM=15 \end{gathered}[/tex]
Now we calculate ND. As OE is a perpendicular bisector of CD, we have:
[tex]ND=\frac{CD}{2}=\frac{30}{2}=15[/tex]
The segment OD is a radius so it has a length like OE. Then, OD = 25.
For the segment ON we can use the Pythagorean theorem again:
[tex]\begin{gathered} ND^2+ON^2=OD^2 \\ ON^2=OD^2-ND^2 \\ ON^2=25^2-15^2 \\ ON^2=625-225 \\ ON^2=400 \\ ON=\sqrt{400} \\ ON=20 \end{gathered}[/tex]
The last segment is MN which length can be expressed as the difference of the lengths of ON and OM:
[tex]MN=ON-OM=20-15=5[/tex]
Answer:
MB = 20
OB = 25
OM = 15
ND = 15
OD = 25
ON = 20
MN = 5
