Final answer:
The true statements are: the central angle created by arc LM is 40°; the ratio of the angle to the circle's total measure is 1/9; Circle O can be divided into 9 equal sectors based on arc LM's measure.
Step-by-step explanation:
If arc LM on circle O has a measure of 40°, let's evaluate the true statements about central angle measures and sector areas.
The relationship between an arc and the central angle of a circle is direct, meaning that the angle measure is equal to the arc it intercepts. Therefore, the central angle measure created by the shaded region is 40°, because it corresponds with the arc LM.
As a full circle is 360°, dividing the central angle measure by the full circle's measure gives a ratio of the angle to the whole circle. Therefore, the ratio of the measure of ∠LOM to the measure of the full circle would be 40/360, which simplifies to 1/9.
Regarding the division of the circle into sectors of equal area, since 360° divided by 40° is 9, circle O can indeed be divided into a total of 9 sectors equal in area to sector LOM. It is not possible to evenly divide the circle into 10 sectors of 40° since 10 sectors would each require an angle of 36°.
Based on these considerations, we can confirm the following statements as true:
The central angle measure created by the shaded region is 40°.
The ratio of the measure of ∠LOM to the measure of the whole circle is 1/9.
Circle O can be divided into a total of 9 sectors equal in area to sector LOM.