G is a circle with chord AB. P is a point outside of G such that PA is tangent to G and ?BPA=90°. If AB=48 and PB=8, what is the radius of G?
To find the radius of circle G, we can use the property of a tangent line to a circle.
Given that PA is tangent to circle G, we know that the radius of G (which we'll call r) is perpendicular to PA at point A.
Let's use the given information to solve for the radius of G:
Given:
AB = 48 (length of the chord)
PB = 8 (length of the line segment from P to B)
Since PA is tangent to the circle, we can form a right triangle PAB where PA is the hypotenuse, and AB is the base.
The length of PA can be found using the Pythagorean theorem:
PA^2 = PB^2 + AB^2 (as PA is the hypotenuse)
PA^2 = 8^2 + 48^2
PA^2 = 64 + 2304
PA^2 = 2368
To find the radius of G, we need to find PA. Taking the square root of both sides:
PA = sqrt(2368)
Now, we know that PA is tangent to the circle and PB is a segment from a point outside the circle to the point of tangency.
Since PB is perpendicular to the radius at point B, it also bisects the chord AB. Therefore, the length of PB is half the length of AB.
PB = AB/2
8 = 48/2
8 = 24
Since PB is equal to 24, and the radius is perpendicular to PB at point B, we have the following right triangle: PBR where RB is the hypotenuse, PB is the base, and the radius of G (r) is the height.
Using the Pythagorean theorem:
RB^2 = PB^2 + BR^2
RB^2 = 24^2 + r^2
But we know that RB = PA - r:
(RB = PA - r)^2 = 24^2 + r^2
(PA - r)^2 = 24^2 + r^2
PA^2 - 2PAr + r^2 = 24^2 + r^2
PA^2 - 24^2 = 2PAr
r = (PA^2 - 576) / 2PA
Substituting the value of PA^2 from earlier:
r = (2368 - 576)/(2 * sqrt(2368))
Evaluating this expression will give us the radius of circle G.