Question

Geometry Problem. Please help I don't understand this question! The diameter AC of circle O is 24 cm. Arcs AB and CD are congruent. Angle DOC = 60° Use this information, the diagram, and your rules for chords, arcs and angles in circles from Unit 4 to solve the questions below. Explain your reasoning for each answer to get full credit. Each one is worth 3 points. Measure of Arc DC = Measure of Arc AB = Measure of Angle ACB = Measure of Angle AOD = Measure of Angle ABC = Measure of Arc BC =

Answer 1

120

Explanation:

Answer 2

A.
`\text{Measure of arc DC}=60^o`

B.
`\text{Measure of arc AB}=60^o`

C.
`m\angle ACB=30^o`

D.
`m\angle AOD=120^o`

E.
`m\angle ABC=90^o`

F.
`\text{Measure of Arc BC}=120^o`

Explanation:

We have been given that the diameter AC of circle O is 24 cm. Arcs AB and CD are congruent. Angle DOC = 60° .

A. Since we know that the degree measure of an arc is equal to the measure of the central angle that intercepts the arc.

We can see that angle DOC is central angle and
`m\angle DOC=60^o` and arc DC subtends to central angle, therefore, the measure of arc DC will be 60 degrees.

B. Since we are told that arcs AB and CD are congruent and measure of arc DC is 60 degrees, therefore, measure of arc AB will be 60 degrees as well.

C. The Inscribed Angle Theorem states that the measure of an inscribed angle is half the measure of its intercepted arc. Measure of Arc AB is 60 degrees, so by inscribed angle theorem the measure of angle ACB will be half the measure of AB.

`m\angle ACB=(1)/(2)* \text{The measure of arc AB}`

`m\angle ACB=(1)/(2)* 60^o`

`m\angle ACB=30^o`

Therefore, measure of angle ACB will be 30 degrees.

D. We can see that angle DOC and AOD form a linear pair of angles, so they will add up-to 180 degrees.

`m\angle DOC+m\angle AOD=180^o`

Upon substituting measure of angle DOC we will get,

`60^o+m\angle AOD=180^o`

`60^o-60^o+m\angle AOD=180^o-60^o`

`m\angle AOD=120^o`

Therefore, measure of angle AOD will be 120 degrees.

E. Since we know that angle inscribed in a semicircle is always a right angle. We can see that angle ABC is angle inscribed in the semicircle as AC is the diameter of our given circle, therefore, measure of angle ABC will be 90 degrees.

F. Since we know that arc measure of a semicircle is 180 degrees. We can see that arc ABC is arc of the semicircle, so we can set an equation as:

`\text{Measure of arc AB+ Measure of Arc BC}=180^o`

Upon substituting measure of arc AB we will get,

`60^o+\text{Measure of Arc BC}=180^o`

`60^o-60^o+\text{Measure of Arc BC}=180^o-60^o`

`\text{Measure of Arc BC}=120^o`

Therefore, measure of arc BC is 120 degrees.