Circle G with center O is tangential to PA at A . Line PO extended intersects G at B (PO<PB ). C is a point on AB such that PC bisects angle APB .What is the measure (in degrees) of angle PCA ?
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To find the measure of angle PCA, we first need to understand the given information and the relationships between the different elements in the figure.
1. Circle G with center O is tangential to PA at A: This means that the line segment PA is tangent to the circle G at point A. In other words, PA is perpendicular to the radius of the circle drawn from center O to point A.
2. Line PO extended intersects G at B: This means that the line segment PO extends beyond the center of the circle and intersects the circle at point B.
3. PC bisects angle APB: This means that the line segment PC divides angle APB into two equal angles.
Now, let's use this information to find the measure of angle PCA.
Since circle G is tangential to PA at A, we know that angle PAB is a right angle (because a tangent to a circle is perpendicular to the radius drawn to the point of tangency). Therefore, angle PAB measures 90 degrees.
Since PC bisects angle APB, angles PBC and PBA are equal. Let's denote the measure of these angles as x degrees.
Since the sum of angles in a triangle is 180 degrees, we can find the measure of angle PBC as follows:
Angle PBC = 180 - (90 + x) degrees
Now, we can find the measure of angle PCA by subtracting the measures of angles PBC and PAB from 180 degrees:
Angle PCA = 180 - (90 + x + (180 - (90 + x))) degrees
Simplifying further:
Angle PCA = 180 - 90 - x - 180 + 90 + x
Angle PCA = 180 - 90 - 180 + 90
Angle PCA = 0 degrees
Therefore, the measure of angle PCA is 0 degrees.