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

Question

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

KnowledgeBoat · Accepted Answer

By formula, By pythagoras theorem, ⇒ (Hypotenuse)² = (Perpendicular)² + Base² In right angle triangle ACD, By pythagoras theorem, ⇒ CD² = AC² + AD² ⇒ CD² = AC² + (AB + BD)² ⇒ CD² = AC² + AB² + BD² + 2.AB.BD .........(1) In right angle triangle ABC, By pythagoras theorem, ⇒ BC² = AC² + AB² ⇒ BC² = AB² + AB² (Since, CA = AB) ⇒ BC² = 2AB² ⇒ AB² = BC² .........(2) Substituting value of AB² from equation (2) in (1), we get : ⇒ CD² = AC² + + BD² + 2.AB.BD ⇒ CD² - BD² = AB² + + 2.AB.BD [As, AC = AB] ⇒ CD² - BD² = AB² + AB² + 2.AB.(AD - AB) [From equation (2)] ⇒ CD² - BD² = 2AB² + 2.AB.AD - 2.AB² ⇒ CD² - BD² = 2.AB.AD Hence, proved that CD² - BD² = 2.AB.AD

Mathematics

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

Pythagoras Theorem

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Mathematics

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

Pythagoras Theorem

Related Questions

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In an isosceles triangle ABC; AB = AC and D is a point on BC produced. Prove that :AD2 = AC2 + BD.CD

In triangle ABC, AB = AC and BD is perpendicular to AC. Prove that :BD2 - CD2 = 2CD × AD

In the following figure, AD is perpendicular to BC and D divides BC in the ratio 1 : 3.Prove that : 2AC2 = 2AB2 + BC2.

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

In the figure, AB = BC and AD is perpendicular to CD. Prove that :
AC² = 2.BC.DC.

In an isosceles triangle ABC; AB = AC and D is a point on BC produced. Prove that :
AD² = AC² + BD.CD

In triangle ABC, AB = AC and BD is perpendicular to AC. Prove that :
BD² - CD² = 2CD × AD

In the following figure, AD is perpendicular to BC and D divides BC in the ratio 1 : 3.
Prove that : 2AC² = 2AB² + BC².