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Explanation:
Check out the diagram below. I recreated the diagram in your textbook, but I added on a few other points and segments.
Point E is the midpoint of segment AB. Segment OE is perpendicular to AB.
Point F is the midpoint of CD. Segment OF is perpendicular to CD.
Right triangle AOE forms with OA = 10 as the hypotenuse (radius of the circle) and AE = 8, since E cuts AB in half.
Use the pythagorean theorem to find that...
a^2+b^2 = c^2
(AE)^2 + (OE)^2 = (OA)^2
8^2 + (OE)^2 = 10^2
64 + (OE)^2 = 100
(OE)^2 = 100-64
(OE)^2 = 36
OE = sqrt(36)
OE = 6
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Because chord CD is the same length as chord AB, this means the chords are the same distance from the center point O.
So, OE = OF = 6
Since we have rectangle OEPF, the opposite sides are congruent and we can say OF = EP = 6
Now focus on right triangle OEP. Use the pythagorean theorem again
a^2+b^2 = c^2
(OE)^2 + (EP)^2 = (OP)^2
(6)^2 + (6)^2 = (OP)^2
72 = (OP)^2
(OP)^2 = 72
OP = sqrt(72)
OP = sqrt(36*2)
OP = sqrt(36)*sqrt(2)
OP = 6*sqrt(2)
