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?
81
PT^2 = PA * PB
PT = 45
PA = 25
PB = (45 * 45)/25
= 81 units
To find the length of PB, we can use a property of tangents to a circle.
Let's draw the given scenario:
```
P
/|\
/ | \
/ | \
/ | \
T | \
Γ
```
From the given information, we have PT = 45 and PA = 25. Let's label the length of PB as x.
Now, let's observe the property of a tangent from a point to a circle: a tangent is perpendicular to the radius drawn to the point of tangency.
This means that triangle PTA is a right triangle, where angle PTB is the right angle.
Using the Pythagorean theorem, we can find the length of the third side, TB:
PT^2 + PB^2 = TB^2
Substituting the given values:
45^2 + x^2 = TB^2
2025 + x^2 = TB^2
Since we know PA = 25 and PT = 45, we can find the length of TA:
TA = PA - PT
TA = 25 - 45
TA = -20
However, note that distance cannot be negative. This implies that point A is on the other side of the circle Γ from point P.
Now, let's consider the triangle PBA:
Using the triangle inequality, we can write:
PA + PB > BA
25 + x > BA
Solving for BA:
BA < 25 + x
Since PA = 25, it means BA cannot be equal to 25 + x. So, BA must be less than 25 + x.
However, we already established that point A is on the other side of the circle Γ, which means BA > 0.
Combining these inequalities:
0 < BA < 25 + x
This implies that BA is strictly greater than 0 and strictly less than 25 + x.
We use this result to find the length of PB.
Now, let's find the length of BA. Since BA is an external secant segment, we can apply the secant-tangent theorem:
PA * PB = PT^2
25 * x = 45^2
25x = 2025
x = 2025/25
x = 81
Therefore, the length of PB is 81.