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In the Figure, Chords Ae and Bc Intersect Each Other at Point D. (I) If` `∠`Cde = 90°, Ab = 5 Cm, Bd = 4 Cm and Cd = 9 Cm Find De. (Ii) If Ad = Bd, Show that Ae = Bc - ICSE Class 10 - Mathematics

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ConceptChord Properties - a Straight Line Drawn from the Center of a Circle to Bisect a Chord Which is Not a Diameter is at Right Angles to the Chord

Question

In the figure, chords AE and BC intersect each other at point D.
(i) If` `∠`CDE = 90°,
AB = 5 cm,
BD = 4 cm and
CD = 9 cm
Find DE.
(ii) If AD = BD, show that AE = BC

Solution

Join AB.
i) In Rt. ΔADB,
 `AB ^2 = AD^2  + DB^2`
`5^2 = AD ^2 + 4^2`
`AD^2 =  25 - 16`
`AD^2 9`
AD = 3
Chords AE and CB intersect each other at D inside the circle
AD x DE = BD x DC
3 × DE = 4 × 9
DE = 12 cm
ii) If AD = BD .......(i)
We know that:
AD × DE = BD × DC
But AD = BD
Therefore, DE = DC .......(ii)
Adding (i) and (ii)
AD + DE = BD + DC
Therefore, AE = BC

 

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Solution In the Figure, Chords Ae and Bc Intersect Each Other at Point D. (I) If` `∠`Cde = 90°, Ab = 5 Cm, Bd = 4 Cm and Cd = 9 Cm Find De. (Ii) If Ad = Bd, Show that Ae = Bc Concept: Chord Properties - a Straight Line Drawn from the Center of a Circle to Bisect a Chord Which is Not a Diameter is at Right Angles to the Chord.
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