Question

Points A and B split the circle into two arcs. Measure of minor arc is 150°. Point M splits major arc with the ratio 2:5 (point M is closer to point B). Find m?BAM.

Answer 1

arc BAM = 300°

Explanation:

Refer to the figure attached.

arc AB = 150°

then:

arc AMB = 360° - 150° = 210°

arc AMB can be decomposed as:

arc AM + arc MB = arc AMB (eq. 1)

From data:

arc MB/arc AM = 2/5

arc MB*(5/2) = arc AM (eq. 2)

Replacing equation 2 into equation 1:

arc MB*(5/2) + arc MB = 210°

arc MB*(7/2) = 210°

arc MB = (2/7)*210° = 60°

On the other hand:

arc MB + arc BAM = 360°

arc BAM = 360° - 60° = 300°

Answer 2

∠BAM=30°

Explanation:

Points A and B split the circle into two arcs. If the measure of minor arc AB is 150°, then the measure of the major arc AB is 360°-150°=210°.

If point M splits major arc with the ratio 2:5, then

• ∠BOM=2x°;
• ∠AOM=5x°.

Angles BOM and AOM together form angle with the measure 210°, thus

2x+5x=210,

7x=210,

x=30°

and ∠BOM=60°, ∠AOM=150°.

Consider isosceles triangle BOM. In this triangle,

∠OBM=1/2(180°-60°)=60°.

Consider isosceles triangle AOB. In this triangle,

∠OAB=1/2(180°-150°)=15°.

Consider isosceles triangle AOM. In this triangle,

∠OAM=1/2(180°-150°)=15°.

Thus, ∠BAM=15°+15°=30°