In △ABC, ∠B = 90° and D is the mid-point of BC. Prove that
(i) AC² = AD² + 3 CD²
(ii) BC² = 4 (AD² - AB²)

Question

In △ABC, ∠B = 90° and D is the mid-point of BC. Prove that (i) AC2 = AD2 + 3 CD2 (ii) BC2 = 4 (AD2 - AB2)

KnowledgeBoat · Accepted Answer

(i) Given, ABC is a triangle such that ∠ABC = 90°. D is the mid-point of BC. In △ ABD, using Pythagoras theorem, Hypotenuse2 = Base2 + Height2 ⇒ AD2 = AB2 + BD2 ......(1) Similarly, in △ ABC, ⇒ AC2 = AB2 + BC2 ⇒ AC2 = AB2 + (BD + DC)2 ⇒ AC2 = AB2 + BD2 + DC2 + 2 × BD × DC As D is the midpoint of BC, BD = DC. ⇒ AC2 = AB2 + BD2 + CD2 + 2 × CD × CD ⇒ AC2 = AB2 + BD2 + CD2 + 2CD2 ⇒ AC2 = AB2 + BD2 + 3CD2 Using equation (1), we get ⇒ AC2 = AD2 + 3CD2 Hence, proved that AC2 = AD2 + 3 CD2. (ii) In △ ABD, ⇒ AD2 = AB2 + BD2 BD = Hence, proved that BC2 = 4 (AD2 - AB2).

Mathematics

In △ABC, ∠B = 90° and D is the mid-point of BC. Prove that
(i) AC² = AD² + 3 CD²
(ii) BC² = 4 (AD² - AB²)

Pythagoras Theorem

Related Questions

In a rhombus ABCD, prove that AC² + BD² = 4AB².

Two poles of height 9 m and 14 m stand vertically on a plane ground. If the distance between their feet is 12 m, find the distance between their tops.

In △ABC, if AB = AC and D is a point on BC. Prove that AB² - AD² = BD × CD.

Mathematics

In △ABC, ∠B = 90° and D is the mid-point of BC. Prove that(i) AC2 = AD2 + 3 CD2(ii) BC2 = 4 (AD2 - AB2)

Pythagoras Theorem

Related Questions

In a rhombus ABCD, prove that AC2 + BD2 = 4AB2.

Two poles of height 9 m and 14 m stand vertically on a plane ground. If the distance between their feet is 12 m, find the distance between their tops.

In △ABC, if AB = AC and D is a point on BC. Prove that AB2 - AD2 = BD × CD.

In △ABC, ∠B = 90° and D is the mid-point of BC. Prove that
(i) AC² = AD² + 3 CD²
(ii) BC² = 4 (AD² - AB²)

In a rhombus ABCD, prove that AC² + BD² = 4AB².

In △ABC, if AB = AC and D is a point on BC. Prove that AB² - AD² = BD × CD.