In Triangle Abc, Angle a = 90o, Ca = Ab and D is the Point on Ab Produced. Prove that Dc2 - Bd2 = 2ab.Ad. - Mathematics

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Sum

In triangle ABC, angle A = 90o, CA = AB and D is the point on AB produced.
Prove that DC2 - BD2 = 2AB.AD.

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Solution

Pythagoras theorem states that in a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the remaining two sides.

We consider the rt. angled  ΔACD and applying Pythagoras theorem we get,
CD2 = AC2 + AD2
CD2 = AC2 + ( AB + BD )2                    ....[ ∵ AD = AB + BD ]
CD2 = AC2 + AB2 + BD2 + 2AB.BD      ...(i)

Similarly, in ΔABC,
BC2 = AC2 + AB2
BC2 = 2AB2                                        ...[ AB = AC ]
AB2 = `1/2`BC                                  ...(ii)

Putting, AB2 from (ii) in (i), We get,
CD2 = AC2 + `1/2`BC2 + BD2 + 2AB . BD

CD2 - BD2 = AB2 + AB2 + 2AB . ( AD - AB )

CD2 - BD2 = AB2 + AB2 + 2AB . AD - 2AB

CD2 - BD2 = 2AB . AD

DC2 - BD2 = 2AB . AD                       

Hence Proved.

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Chapter 13: Pythagoras Theorem [Proof and Simple Applications with Converse] - Exercise 13 (B) [Page 164]

APPEARS IN

Selina Concise Mathematics Class 9 ICSE
Chapter 13 Pythagoras Theorem [Proof and Simple Applications with Converse]
Exercise 13 (B) | Q 13 | Page 164

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