A point P is given outside of a circle Γ. A tangent from P to Γ touches Γ at T with PT=45. A line from P cuts Γ at the 2 points A,B. If PA=25, what is the length of PB?
45*45=25*x
x=81
Thanks hans!
To find the length of PB, we can use the property of tangent lines from an external point to a circle.
First, let's draw a diagram to better understand the situation:
```
A
/|
/ |
/ |
/ |
/ | 25
T / |
------
45
------
P
```
From the diagram, we can see that PT is the tangent line from point P to the circle Γ. We are given that PT = 45.
According to the property of tangent lines, the tangent and radius of a circle are perpendicular to each other at the point of tangency (in this case, point T). Therefore, angle PT T is a right angle.
Next, we draw a line from point P to point B and extend it to intersect the circle at point X.
```
A
/|
/ |
/ |
/ |
/ X | 25
T / |
------
45
------
P
```
Since the tangent PT is perpendicular to the radius, angle PT T is also a right angle. Therefore, angle PT B is a right angle as well because it is formed by a line passing through points P and X and the tangent PT.
Now, we have a right triangle PT B. We are given that PT = 45 and PA = 25. We need to find the length of PB.
Using the Pythagorean theorem, we can find the length of PB:
(TB)^2 + (PT)^2 = (PB)^2
Substituting the given values:
(TB)^2 + (45)^2 = (PB)^2
To find the length of TB, we need to subtract PA from PT:
TB = PT - PA = 45 - 25 = 20
Substituting this value into the equation:
(20)^2 + (45)^2 = (PB)^2
Simplifying:
400 + 2025 = (PB)^2
2425 = (PB)^2
Taking the square root of both sides to solve for PB:
PB = sqrt(2425)
Therefore, the length of PB is approximately sqrt(2425) units.