Question

arc lm on circle o has a measure of 40°. which statements are true? check all that apply. the central angle measure created by the shaded region is 40°. the central angle measure created by the shaded region is 20°. the ratio of the measure of ∠lom to the measure of the whole circle is \\(\frac{1}{9}\\). circle o can be divided into a total of 9 sectors equal in area to sector lom. circle o can be divided into a total of 10 sectors equal in area to sector lom.

Explanation:

Step1: Analyze central angle of arc LM

The measure of a central angle is equal to the measure of its intercepted arc. Arc LM is \(40^\circ\), so \(\angle LOM = 40^\circ\). Thus, the central angle of the shaded region (sector LOM) is \(40^\circ\), so the first statement is true, the second is false.

Step2: Ratio of \(\angle LOM\) to full circle

A full circle is \(360^\circ\). The ratio is \(\frac{40^\circ}{360^\circ}=\frac{1}{9}\), so the third statement is true.

Step3: Number of equal sectors

To find how many sectors equal to LOM fit in the circle, divide \(360^\circ\) by \(40^\circ\): \(\frac{360}{40} = 9\). So the circle can be divided into 9 equal - area sectors like LOM, so the fourth statement is true, the fifth is false.

Answer:

The true statements are:

• The central angle measure created by the shaded region is \(40^\circ\).
• The ratio of the measure of \(\angle LOM\) to the measure of the whole circle is \(\frac{1}{9}\).
• Circle O can be divided into a total of 9 sectors equal in area to sector LOM.