In the given figure of circle, if RT = 90, SU = 25, and SO is perpendicular to RT, the length of the radius RO is 53.
In the given figure,
RT = 90
SU = 25
SO is perpendicular to RT.
Here, SO and RO are the radii of the circle and RT is a chord.
The length of the chord, RT is given as,
RT = 2 √(r² - d²)
Here, r is the radius and d is the distance of the chord RT from the center O.
Radius, r = RO and distance, d = OU
∴ RT = 2 √(RO² - OU²)
Substituting RT = 90 in the above equation, we get,
90 = 2 √[(RO)² - (OU)²]
√[RO)² - (OU)²] = 45
(RO)² - (OU)² = 45²
(RO)² - (OU)² = 2025 ........... (1)
Now, OU = SO - SU [From the figure]
⇒ OU = SO - 25
Substituting OU = SO - 25 in equation (1), we obtain,
(RO)² - (SO - 25)² = 2025
(RO)²- [(SO)² + (25)² - 2(SO)(25)] = 2025 [ ∵ (a-b)² = a²+b²-2ab ]
(RO)²- (SO)² - (25)² + 50(SO) = 2025 ........... (2)
Since, RO and SO both are the radii of the same circle, we have,
SO = RO
Thus, we can write equation (2), as follows,
⇒ (RO)² - (RO)² - (25)² + 50(RO) = 2025
⇒ -625 + 50RO = 2025
⇒ 50RO = 2025 + 625
⇒ RO = 2650/50
⇒ RO = 53
Hence, the length of the radius RO of the given circle is 53.
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