an inscribed angle with a diameter as a side has a measure of x degrees. if the ratio of measure of arc AD to measure of arc DB is 1:4, what is the measure of arc DB?
arcAD:arcDB = 1:4 means
m<A : m<B = 1:4
Since AB is a diameter, m<D = x = 90
that should help.
To solve this problem, we need to understand the relationship between an inscribed angle and the corresponding arc in a circle.
In a circle, an inscribed angle is an angle whose vertex lies on the circle, and its sides are chords of the circle. The measure of an inscribed angle is half the measure of its intercepted arc.
Given that the inscribed angle has the diameter as one of its sides, we can conclude that the intercepted arc is the entire circumference of the circle, making it 360 degrees.
Let's assume that the measure of the inscribed angle is x degrees. Since the measure of the inscribed angle is half the measure of the intercepted arc, we have:
x = (1/2) * 360
x = 180 degrees
Now, let's consider the ratio of the measure of arc AD to the measure of arc DB, which is given as 1:4. We know that the sum of these two arcs must be 360 degrees.
Let's assume the measure of arc AD is y degrees. Then the measure of arc DB will be 4y, according to the given ratio.
So, we can set up the equation:
y + 4y = 360
Combining like terms, we get:
5y = 360
To solve for y, we divide both sides of the equation by 5:
y = 360 / 5
y = 72 degrees
Finally, we can find the measure of arc DB, which is 4y:
arc DB = 4 * 72
arc DB = 288 degrees
Therefore, the measure of arc DB is 288 degrees.