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

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

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

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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

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

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

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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

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.

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

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.

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.