To find the measure of angle YAC (denoted as yAC), we start by reviewing some important properties of circles and tangents. When a tangent line intersects a circle, it forms a right angle with the radius that connects the point of tangency to the center of the circle.
In this scenario, we have:
- The circle centered at point O.
- Line AC as the tangent at point A.
- The measure of angle BY is given as 34 degrees.
According to the properties of tangents and circles:
- The angle formed between the tangent (AC) and the radius (OA) at point A is a right angle, or 90 degrees.
- The angles around point A should also satisfy the rule that the sum of angles in a triangle is equal to 180 degrees.
Since angle AOY (the angle at the center of the circle from point O to points A and Y) consists of angles YAC and B, we know:
Angle AOY = Angle YAC + Angle AOB
We also know that Angle AOB must be equal to angle AOB because it is subtended by the same arc that angle BY subtends. Therefore:
Angle AOB = 2 * Angle BY = 2 * 34 degrees = 68 degrees.
Now we can apply the right triangle rule at point A:
Angle YAC + 68 degrees + 90 degrees = 180 degrees
Solving for angle YAC, we get:
Angle YAC = 180 degrees – 68 degrees – 90 degrees
Angle YAC = 22 degrees.
Thus, the measure of angle YAC is 22 degrees.