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In the circle below, chord \(\overline {AB}\) is extended to meet the tangent \(\overline {DE}\) at D. If \(\overline {AB}\) = 24 cm \(\overline {DE}\) = 9 cm, what is the length of \(\overline {BD}\)?

In the circle below, chord \(\overline {AB}\) is extended to meet the tangent \(\overline {DE}\) at D. If \(\overline {AB}\) = 24 cm \(\overline {DE}\) = 9 cm, what is the length of \(\overline {BD}\)? Correct Answer 3 cm

Let, length of BD be x cm

According to the question,

BD × AD = DE2

⇒ BD × (BD + AB) = DE2

⇒ x × (x + 24) = 92

⇒ x2 + 24x – 81 = 0

⇒ x2 + 27x – 3x – 81 = 0

⇒ x(x + 27) – 3(x + 27) = 0

⇒ (x + 27) (x – 3) = 0

⇒ x = 3 or – 27

∴ Length of BD = 3 cm

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