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 ?

38 cm

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.