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Answer: Choice B. 10+4*sqrt(14)</h3>
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Explanation:
The diagram shows that PT = 4 and QR = 5. Since QR is a radius of the circle, this means that the other radii segments TR and RS are also 5 units long.
Note how PR = PT+TR = 4+5 = 9, and how this segment is the hypotenuse for the right triangles PQR and PRS.
Picking either triangle, we have one known leg of 5 units (the radius mentioned earlier) and the other is unknown. The unknown leg is tangent to the circle, and recall that a tangent segment is always perpendicular to the radius at the point of tangency. So that's why we have right triangles.
Apply the pythagorean theorem to find the length of PQ and PS (both are the same length)
a^2 + b^2 = c^2
a^2 + 5^2 = 9^2
a^2 + 25 = 81
a^2 = 81 - 25
a^2 = 56
a = sqrt(56)
a = sqrt(4*14)
a = sqrt(4)*sqrt(14)
a = 2*sqrt(14)
The exact length of PQ and PS is 2*sqrt(14) units.
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We have enough to find the perimeter of quadrilateral PQRS. We add up the exterior sides. That means we ignore the interior segment PR.
perimeter = PQ+QR+RS+SP
perimeter = 2*sqrt(14)+5+5+2*sqrt(14)
perimeter = 10+4*sqrt(14)
Answer is choice B
Side note: The expression above approximates to roughly 24.966629
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