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

Draw AP ⊥ BC. APD is right triangle By pythagoras theorem we get, ⇒ AD 2 = AP 2 + PD 2 ⇒ AD 2 = AP 2 + (PC + CD) 2 ⇒ AD 2 = AP 2 + PC 2 + CD 2 + 2PC.CD ......(i) APC is right triangle, By pythagoras theorem we get, ⇒ AC 2 = AP 2 + PC 2 ......(ii) Substituting the value of AP 2 + PC 2 from (ii) in (i) we get, ⇒ AD 2 = AC 2 + CD 2 + 2PC.CD .......(iii) Since, ABC is an isosceles triangle, ⇒ PC = [∵ altitude to the base of an isosceles triangle bisects the base] ⇒ AD 2 = AC 2 + CD 2 + ⇒ AD 2 = AC 2 + CD 2 + BC.CD ⇒ AD 2 = AC 2 + CD(CD + BC) From figure, CD + BC = BD ⇒ AD 2 = AC 2 + CD.BD Hence, proved that AD 2 = AC 2 + BD × CD.

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

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In the adjoining figure, the diagonals AC and BD of a quadrilateral ABCD intersect at O, at right angles. Prove that AB² + CD² = AD² + BC².

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In a quadrilateral ABCD, ∠B = 90° = ∠D. Prove that
2AC² - BC² = AB² + AD² + DC².

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If AD, BE and CF are medians of △ABC, prove that
3(AB² + BC² + CA²) = 4(AD² + BE² + CF²).

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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 a △ABC, ∠A = 90°, CA = AB and D is a point on AB produced. Prove that
DC² - BD² = 2AB × AD.

In the adjoining figure, the diagonals AC and BD of a quadrilateral ABCD intersect at O, at right angles. Prove that AB² + CD² = AD² + BC².

In a quadrilateral ABCD, ∠B = 90° = ∠D. Prove that
2AC² - BC² = AB² + AD² + DC².

If AD, BE and CF are medians of △ABC, prove that
3(AB² + BC² + CA²) = 4(AD² + BE² + CF²).

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 a △ABC, ∠A = 90°, CA = AB and D is a point on AB produced. Prove thatDC2 - BD2 = 2AB × AD.

In the adjoining figure, the diagonals AC and BD of a quadrilateral ABCD intersect at O, at right angles. Prove that AB2 + CD2 = AD2 + BC2.

In a quadrilateral ABCD, ∠B = 90° = ∠D. Prove that2AC2 - BC2 = AB2 + AD2 + DC2.

If AD, BE and CF are medians of △ABC, prove that3(AB2 + BC2 + CA2) = 4(AD2 + BE2 + CF2).

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

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

In the adjoining figure, the diagonals AC and BD of a quadrilateral ABCD intersect at O, at right angles. Prove that AB² + CD² = AD² + BC².

In a quadrilateral ABCD, ∠B = 90° = ∠D. Prove that
2AC² - BC² = AB² + AD² + DC².

If AD, BE and CF are medians of △ABC, prove that
3(AB² + BC² + CA²) = 4(AD² + BE² + CF²).