The radii of two concentric circles with centre O are 26 cm and 16 cm. Chord AB of the larger circle is tangent to the smaller circle at C and AD is a diameter. What is the length of CD ?
Answer & explanation
Correct answer: option 4
According to the question,
OA = OD = 26 cm
OC = 16 cm
If a line divides any two sides in the same ratio (\(\frac{AO}{OD}\) = \(\frac{AC}{CB}\)) then the line is parallel to the third line.
DB parallel to OC
BD = 2OC
BD = 2 x OC
⇒ BD = 2 x 16 = 32 cm.
In \(\Delta \)AOC
⇒ \(\frac{OA}{2}\) = \(\frac{OC}{2}\) + \(\frac{AC}{2}\)
⇒ \(\frac{26}{2}\) = \(\frac{16}{2}\) + \(\frac{AC}{2}\)
⇒ AC = \(\sqrt {420 }\) cm
⇒ AC = BC = \(\sqrt {420 }\) cm
In \(\Delta \)BCD
⇒ \(\frac{CD}{2}\) = \(\frac{BD}{2}\) + \(\frac{BC}{2}\)
⇒ \(\frac{CD}{2}\) = \(\frac{32}{2}\) + \(\frac{√420}{2}\)
⇒ \(\frac{CD}{2}\) = 1024 + 420
⇒ CD = \(\sqrt {1444}\)
⇒ CD =38 cm
Therefore, the length of CD is 38 cm.