Answer:
EK = 1.
Step-by-step explanation:
Here we have KP = PE (tangents from outside the circle)
Therefore, 2KP = PE + 1 = KP + 1
Hence, 2KP - KP = 1 or KP = 1 = PE
Since KE is the base of triangle KPE, where ∡ KPE = 60, and KP = PE, we have an isosceles triangle such that ∡PKE = ∡PEK
Hence, in ΔKPE, ∡KPE + ∡PKE + ∡PEK = 180
Therefore, 60° + ∡PKE + ∡PEK = 180
Hence, ∡PKE + ∡PEK = 180° - 60° = 120°
Because ∡PKE = ∡PEK, (base angles of isosceles triangle), we have;
∡PKE + ∡PEK = 2·∡PEK = 120° which gives
∡PEK = 60° = ∡PKE
Therefore, ∡KPE = ∡PEK = ∡PKE = 60°
Hence, ΔKPE is an equilateral triangle and KP = PE = EK = 1
EK = 1.
