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

Given, D divides BC in the ratio 1 : 3. ⇒ BD : DC = 1 : 3 Let BD = x and DC = 3x From figure, ⇒ BC = BD + DC = x + 3x = 4x. In right angle triangle ADC, By pythagoras theorem, ⇒ AC 2 = AD 2 + CD 2 ............(1) In right angle triangle ABD, By pythagoras theorem, ⇒ AB 2 = AD 2 + BD 2 ............(2) Subtracting equation (2) from (1), we get : ⇒ AC 2 - AB 2 = AD 2 + CD 2 - (AD 2 + BD 2 ) ⇒ AC 2 - AB 2 = CD 2 - BD 2 Substituting value of BD and CD in above equation, we get : Hence, proved that 2AC 2 = 2AB 2 + BC 2 .

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

View Answer Bookmark Now

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

View Answer Bookmark Now

In the given figure, AB = 16 cm, BC = 12 cm and CA = 6 cm; find the length of CD.

View Answer Bookmark Now

In a quadrilateral ABCD, ∠A + ∠D = 90°, prove that : AC² + BD² = AD² + BC².

View Answer Bookmark Now

Mathematics

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

Pythagoras Theorem

Related Questions

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 the given figure, AB = 16 cm, BC = 12 cm and CA = 6 cm; find the length of CD.

In a quadrilateral ABCD, ∠A + ∠D = 90°, prove that : AC² + BD² = AD² + BC².

Mathematics

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

Pythagoras Theorem

Related Questions

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 the given figure, AB = 16 cm, BC = 12 cm and CA = 6 cm; find the length of CD.

In a quadrilateral ABCD, ∠A + ∠D = 90°, prove that : AC2 + BD2 = AD2 + BC2.

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

In a quadrilateral ABCD, ∠A + ∠D = 90°, prove that : AC² + BD² = AD² + BC².