In the given figure, △ABC is an equilateral triangle and BC is produced to D such that BC = CD. Prove that AD ⊥ AB.

Given, △ABC is an equilateral triangle. ⇒ ∠ABC = ∠ACB = ∠BAC = 60° From figure, ⇒ ∠ACB + ∠ACD = 180° (Linear pair) ⇒ 60° + ∠ACD = 180° ⇒ ∠ACD = 180° - 60° ⇒ ∠ACD = 120°. In △CAD, CA = CD (As, BC = CD and BC = CA) ⇒ ∠CAD = ∠CDA = x (let) (Angles opposite to equal sides are equal) By angle sum property of triangle, ⇒ ∠CAD + ∠CDA + ∠ACD = 180° ⇒ x + x + 120° = 180° ⇒ 2x = 180° - 120° ⇒ 2x = 60° ⇒ x = ⇒ x = 30° ⇒ ∠CAD = ∠CDA = 30° From figure, ⇒ ∠BAD = ∠BAC + ∠CAD = 60° + 30° = 90°. Hence, proved that AD ⊥ AB.

In the given figure, AD is the internal bisector of ∠A and CE || DA. If CE meets BA produced at E, prove that △CAE is isosceles.

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In the adjoining figure, AB = AC. If DB ⊥ BC and EC ⊥ BC, prove that :
(i) BD = CE
(ii) AD = AE

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In the given figure, AC is the bisector of ∠A. If AB = AC, AD = CD and ∠ABC = 75°, find the values of x and y.

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In a △PQR, ∠P = 50° and ∠R = 70°. Name :
(i) the shortest side
(ii) the longest side of the triangle

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Mathematics

In the given figure, △ABC is an equilateral triangle and BC is produced to D such that BC = CD. Prove that AD ⊥ AB.

Triangles

Related Questions

In the given figure, AD is the internal bisector of ∠A and CE || DA. If CE meets BA produced at E, prove that △CAE is isosceles.

In the adjoining figure, AB = AC. If DB ⊥ BC and EC ⊥ BC, prove that :
(i) BD = CE
(ii) AD = AE

In the given figure, AC is the bisector of ∠A. If AB = AC, AD = CD and ∠ABC = 75°, find the values of x and y.

In a △PQR, ∠P = 50° and ∠R = 70°. Name :
(i) the shortest side
(ii) the longest side of the triangle

Mathematics

In the given figure, △ABC is an equilateral triangle and BC is produced to D such that BC = CD. Prove that AD ⊥ AB.

Triangles

Related Questions

In the given figure, AD is the internal bisector of ∠A and CE || DA. If CE meets BA produced at E, prove that △CAE is isosceles.

In the adjoining figure, AB = AC. If DB ⊥ BC and EC ⊥ BC, prove that :(i) BD = CE(ii) AD = AE

In the given figure, AC is the bisector of ∠A. If AB = AC, AD = CD and ∠ABC = 75°, find the values of x and y.

In a △PQR, ∠P = 50° and ∠R = 70°. Name :(i) the shortest side(ii) the longest side of the triangle

In the adjoining figure, AB = AC. If DB ⊥ BC and EC ⊥ BC, prove that :
(i) BD = CE
(ii) AD = AE

In a △PQR, ∠P = 50° and ∠R = 70°. Name :
(i) the shortest side
(ii) the longest side of the triangle