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In the figure given below, ABCD is cyclic quadrilateral and AC is the diameter of the circle. BC = 7 cm, CD = 15 cm, AD∶ AB = 5∶ 6, then what is the length of diagonal BD of ABCD?

In the figure given below, ABCD is cyclic quadrilateral and AC is the diameter of the circle. BC = 7 cm, CD = 15 cm, AD∶ AB = 5∶ 6, then what is the length of diagonal BD of ABCD? Correct Answer 20 cm

Concept:

According to Ptolemy's theorem in cyclic quadrilateral∶

AC × BD = AB × CD + BC × AD

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Let AD and AB be ‘5x‘ and ’6x’ respectively and AC is ‘y’.

Since, AC is the diameter of circle. So, ∠ADC = ∠ABC = 90°

In ΔABC∶

AC2 = AB2 + BC2

y2 = (6x)2 + 72

y2 = 36x2 + 49       ----(1)

In ΔADC∶

AC2 = AD2 + DC2

y2 = (5x)2 + 152

y2 = 25x2 + 225       ----(2)

From (1) and (2)∶

36x2 + 49 = 25x2 + 225

x2 = 16

x = 4 and y = 25

AC = y = 25 cm

AB = 6x = 24 cm and AD = 5x = 20 cm

Now,

According to Ptolemy's theorem in cyclic quadrilateral∶

AC × BD = AB × CD + BC × AD

25 × BD = 24 × 15 + 7 × 20

25 × BD = 500

BD = 20 cm

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