Math / Geometry

QuestionTC and WI dimmers a 0 R $\square$ Find ins meame. $\square$

Studdy Solution

STEP 1

What is this asking? We need to figure out if arc DA is a major arc, a minor arc, or a semicircle, and then find how many degrees it measures. Watch out! Don't mix up the different types of arcs!
Remember, a minor arc is less than $180^\circ$ , a major arc is greater than $180^\circ$ , and a semicircle is exactly $180^\circ$ .

STEP 2

1. Find the measure of arc BC.
2. Find the measure of arc DC.
3. Find the measure of arc DAB.
4. Find the measure of arc DA.
5. Classify arc DA.

STEP 3

We are given that $\angle COB$ measures $50^\circ$ .
Since a central angle has the same measure as its intercepted arc, the measure of arc BC is also $50^\circ$ .
So, $m\overset{\frown}{BC} = 50^\circ$ .

STEP 4

We're given that $m\angle DOE = 30^\circ$ and $m\angle EOC = 100^\circ$ .
Since angles DOE and EOC add up to form angle DOC, we can find the measure of angle DOC by adding these two angles together.
So, $m\angle DOC = m\angle DOE + m\angle EOC = 30^\circ + 100^\circ = 130^\circ$ .

STEP 5

Now, since angle DOC is a central angle and arc DC is its intercepted arc, they have the same measure.
Therefore, $m\overset{\frown}{DC} = 130^\circ$ .

STEP 6

We know that DB is a diameter of the circle, which means it divides the circle into two semicircles.
A semicircle measures $180^\circ$ .
Since arc DAB is a semicircle, $m\overset{\frown}{DAB} = 180^\circ$ .

STEP 7

We know that $m\overset{\frown}{DAB} = m\overset{\frown}{DA} + m\overset{\frown}{AB}$ .
We also know that $m\overset{\frown}{DAB} = 180^\circ$ .
Since $AC$ is a diameter, we know that arc ABC is a semicircle, so $m\overset{\frown}{ABC} = 180^\circ$ .

STEP 8

We also know that $m\overset{\frown}{ABC} = m\overset{\frown}{AB} + m\overset{\frown}{BC}$ and $m\overset{\frown}{BC} = 50^\circ$ .
So, $180^\circ = m\overset{\frown}{AB} + 50^\circ$ .
Subtracting $50^\circ$ from both sides gives us $m\overset{\frown}{AB} = 180^\circ - 50^\circ = 130^\circ$ .

STEP 9

Now we can find the measure of arc DA!
We know $m\overset{\frown}{DAB} = m\overset{\frown}{DA} + m\overset{\frown}{AB}$ , so $180^\circ = m\overset{\frown}{DA} + 130^\circ$ .
Subtracting $130^\circ$ from both sides gives us $m\overset{\frown}{DA} = 180^\circ - 130^\circ = 50^\circ$ .

STEP 10

Since $m\overset{\frown}{DA} = 50^\circ$ , and $50^\circ$ is less than $180^\circ$ , arc DA is a minor arc.

STEP 11

Arc DA is a minor arc and its measure is $\bold{50^\circ}$ .

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