pa and pb are tangents to circle from point p. chord ab is drawn. find the number of degrees contained in angle pab if angle apb contains
a)80 degees
b 40 degrees
c) 60 degrees
d) 90 degrees
e) 120 degrees
can some1 pllllz explain this 2 me!
To find the measure of angle PAB, we need to use the properties of tangents and chords in a circle.
First, let's understand the relationship between tangents and radii in a circle. When a tangent intersects a radius of a circle, it forms a right angle at the point of contact.
Let's consider case (a) where angle APB measures 80 degrees:
Since PA and PB are tangents to the circle, angle APB must be half of the central angle that subtends the same arc (in this case, chord AB).
Since angle APB measures 80 degrees, the central angle that subtends chord AB will measure twice that value, which is 160 degrees.
Now, let's find angle PAB. It is an inscribed angle that intercepts the same arc as the central angle. By using the Inscribed Angle Theorem, we know that the measure of an inscribed angle is half the measure of the central angle that subtends the same arc.
Therefore, angle PAB will measure half of the central angle that is 160 degrees. So, angle PAB will measure 80 degrees.
Hence, the answer for case (a) is 80 degrees.
You can use similar reasoning for the other cases as well:
For case (b), angle APB measures 40 degrees. Therefore, the central angle is twice that value, which is 80 degrees. So, angle PAB will measure half the central angle, which is 40 degrees.
For case (c), angle APB measures 60 degrees. Therefore, the central angle is twice that value, which is 120 degrees. So, angle PAB will measure half the central angle, which is 60 degrees.
For case (d), angle APB measures 90 degrees. Therefore, the central angle is twice that value, which is 180 degrees. However, since angle PAB is an inscribed angle, it cannot be larger than a semicircle (which measures 180 degrees), so angle PAB will measure 180 - 90 = 90 degrees.
For case (e), angle APB measures 120 degrees. Therefore, the central angle is twice that value, which is 240 degrees. So, angle PAB will measure half the central angle, which is 120 degrees.
Hence, the measures of angle PAB for each case are:
a) 80 degrees
b) 40 degrees
c) 60 degrees
d) 90 degrees
e) 120 degrees