The endpoints of the arms of an inscribed angle are the endpoints of a diameter. What is the measure of the inscribed angle? How do you know?
Please help and please explain how I would draw a diagram of thus. Think you for your help!
it is a right angle.
The central angle is 180°
An inscribed angle is 1/2 the central angle.
Check your theorems.
To find the measure of an inscribed angle where the endpoints of the arms are the endpoints of a diameter, we can use a key property: an inscribed angle intercepts the same arc as its corresponding central angle.
To understand this, let's start by drawing a circle. Then, draw a diameter passing through the center (let's call it AB). The two endpoints of the diameter (A and B) become the endpoints of the arms of the inscribed angle.
To draw the inscribed angle, choose a point on the circle that is not on the diameter, and connect it to A and B (creating two lines). This forms an angle, and the measure of this angle is what we are trying to find.
Now, draw a line segment from the center of the circle (the point where the diameter intersects) to the point you chose on the circle (let's call it C). This line segment is the radius of the circle and is perpendicular to the diameter.
To find the measure of the inscribed angle, we need to consider the central angle formed by the three points: A, C, and B. This central angle, labeled as ∠ACB, has its vertex at the center of the circle.
Since we know that an inscribed angle and its corresponding central angle intercept the same arc, the measure of the inscribed angle ∠AEB is equal to half the measure of its corresponding central angle ∠ACB.
In other words, if we calculate the measure of ∠ACB, we can divide it by 2 to find the measure of the inscribed angle ∠AEB.
To find the measure of ∠ACB, we can use the fact that an inscribed angle intercepts the same arc as the central angle. The intercepted arc is the arc between points A and B on the circle.
To find the measure of the intercepted arc, we can observe that it is a semicircle (since it is formed by the diameter AB). We know that a semicircle has a measure of 180 degrees.
Therefore, the central angle ∠ACB (and the intercepted arc) has a measure of 180 degrees. Consequently, the measure of the inscribed angle ∠AEB is half of that, which is 90 degrees.
So, the measure of the inscribed angle is 90 degrees. We can be confident of this based on the property that an inscribed angle's measure is half the measure of the central angle that intercepts the same arc.
I hope this explanation helps! Let me know if you have any further questions.