In an isosceles triangle ABC; AB = AC and D is a point on BC produced. Prove that : AD 2 = AC 2 + BD.CD

By formula, By pythagoras theorem, ⇒ (Hypotenuse) 2 = (Perpendicular) 2 + Base 2 Draw AE perpendicular to BC. In right angle triangle AED, By pythagoras theorem, ⇒ AD 2 = AE 2 + ED 2 ⇒ AD 2 = AE 2 + (EC + CD) 2 .........(1) In right angle triangle AEC, By pythagoras theorem, ⇒ AC 2 = AE 2 + EC 2 ⇒ AE 2 = AC 2 - EC 2 .........(2) Substituting value of AE 2 from equation (2) in (1), we get : ⇒ AD 2 = AC 2 - EC 2 + (EC + CD) 2 ⇒ AD 2 = AC 2 - EC 2 + EC 2 + CD 2 + 2.EC.CD ⇒ AD 2 = AC 2 + CD(CD + 2EC) ..........(3) Since, ABC is an isosceles triangle. We know that, In an isosceles triangle altitude from the vertex bisects the base. ∴ E is the mid-point of BC. ⇒ EC = BC ⇒ BC = 2EC. From figure, ⇒ BD = BC + CD ⇒ BD = 2EC + CD Substituting above value in equation (3), we get : ⇒ AD 2 = AC 2 + CD.BD Hence, proved that AD 2 = AC 2 + BD.CD

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

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

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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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Mathematics

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

Pythagoras Theorem

Related Questions

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

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

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

Mathematics

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

Pythagoras Theorem

Related Questions

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

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

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

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

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

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

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