In a quadrilateral ABCD, ∠B = 90° = ∠D. Prove that 2AC 2 - BC 2 = AB 2 + AD 2 + DC 2 .

Given, ∠B = 90° = ∠D By pythagoras theorem, In right angle triangle ABC, AC 2 = AB 2 + BC 2 ......(i) By pythagoras theorem, In right angle triangle ADC, AC 2 = AD 2 + DC 2 ......(ii) Adding (i) and (ii) we get, 2AC 2 = AB 2 + BC 2 + AD 2 + DC 2 2AC 2 - BC 2 = AB 2 + AD 2 + DC 2 . Hence, proved that 2AC 2 - BC 2 = AB 2 + AD 2 + DC 2 .

Mathematics

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

Pythagoras Theorem

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Answer

Given, ∠B = 90° = ∠D

In a quadrilateral ABCD, ∠B = 90° = ∠D. Prove that 2AC^2 - BC^2 = AB^2 + AD^2 + DC^2. Pythagoras Theorem, ML Aggarwal Understanding Mathematics Solutions ICSE Class 9.

By pythagoras theorem,

In right angle triangle ABC,

AC² = AB² + BC² ……(i)

By pythagoras theorem,

In right angle triangle ADC,

AC² = AD² + DC² ……(ii)

Adding (i) and (ii) we get,

2AC² = AB² + BC² + AD² + DC²

2AC² - BC² = AB² + AD² + DC².

Hence, proved that 2AC² - BC² = AB² + AD² + DC².

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Mathematics

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

Pythagoras Theorem

Answer

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Mathematics

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

Pythagoras Theorem

Answer

Related Questions

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

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

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).

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

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