Option A: 4 is the length of PE
Explanation:
Given that PA=6, PD=4, and BE=5
We need to determine the length of PE
The length of PE can be determined using the intersecting secant tangent theorem.
Applying the secant tangent theorem, we get,
[tex]PA^2=(BE+PE)(PE)[/tex]
Substituting the values of the PA and BE, we get,
[tex]6^2=(5+PE)(PE)[/tex]
Simplifying, we get,
[tex]36=5PE+PE^2[/tex]
Subtracting both sides of the equation by 36, we get,
[tex]0=-36+5PE+PE^2[/tex]
Switch sides, we get,
[tex]PE^2+5PE-36=0[/tex]
Solving this equation, we get,
[tex](PE+9)(PE-4)=0[/tex]
Equating each term equal to zero, we get,
[tex]PE+9=0[/tex] and [tex]PE-4=0[/tex]
Simplifying, we get,
[tex]PE=-9[/tex] and [tex]PE=4[/tex]
The value of PE cannot be negative.
Thus, the length of PE is 4.
Hence, Option A is the correct answer.
