Answer: PU = 72
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Step-by-step explanation:
V is the circumcenter of the triangle PQR, which means that it's the center of the circle that goes through points P, Q, and R.
Furthermore it means
PV = QV = RV
as they are all radii of circle V.
We only really need to focus on PV = RV.
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We're given that RV = 78, which then points us to PV = 78 as well.
We're also given that UV = 30.
Segment PU is the missing piece of right triangle PUV.
As you can probably guess, we'll use the pythagorean theorem to find this missing side.
a^2+b^2 = c^2
(PU)^2 + (UV)^2 = (PV)^2
(PU)^2 + (30)^2 = (78)^2
(PU)^2 + 900 = 6084
(PU)^2 = 6084 - 900
(PU)^2 = 5184
PU = sqrt(5184)
PU = 72