AD is perpendicular to the side BC of an equilateral △ABC. Prove that 4AD 2 = 3AB 2 .

Given, AD ⊥ BC and AB = BC = CA (Equilateral triangle). The perpendicular to base in equilateral triangle bisects the base. ∴ BD = . From figure, In right triangle ABD, By pythagoras theorem, ⇒ AB 2 = AD 2 + BD 2 ⇒ AB 2 = AD 2 + ⇒ AB 2 = AD 2 + ⇒ AB 2 = ⇒ 4AB 2 = 4AD 2 + BC 2 ⇒ 4AB 2 = 4AD 2 + AB 2 (∵ BC = AB) ⇒ 4AD 2 = 3AB 2 . Hence, proved that 4AD 2 = 3AB 2 .

In figure given below, AB || DC, ∠A = 90°, DC = 7 cm, AB = 17 cm and AC = 25 cm. Calculate BC.

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In the figure given below, ABCD is a square of side 7 cm. If
AE = FC = CG = HA = 3 cm,
(i) prove that EFGH is a rectangle.
(ii) find the area and perimeter of EFGH.

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In the adjoining figure, D and E are mid-points of the sides BC and CA respectively of a △ABC, right angled at C. Prove that :
(i) 4AD² = 4AC² + BC²
(ii) 4BE² = 4BC² + AC²
(iii) 4(AD² + BE²) = 5AB²

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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

AD is perpendicular to the side BC of an equilateral △ABC. Prove that 4AD² = 3AB².

Pythagoras Theorem

Related Questions

In figure given below, AB || DC, ∠A = 90°, DC = 7 cm, AB = 17 cm and AC = 25 cm. Calculate BC.

In the figure given below, ABCD is a square of side 7 cm. If
AE = FC = CG = HA = 3 cm,
(i) prove that EFGH is a rectangle.
(ii) find the area and perimeter of EFGH.

In the adjoining figure, D and E are mid-points of the sides BC and CA respectively of a △ABC, right angled at C. Prove that :
(i) 4AD² = 4AC² + BC²
(ii) 4BE² = 4BC² + AC²
(iii) 4(AD² + BE²) = 5AB²

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

Mathematics

AD is perpendicular to the side BC of an equilateral △ABC. Prove that 4AD2 = 3AB2.

Pythagoras Theorem

Related Questions

In figure given below, AB || DC, ∠A = 90°, DC = 7 cm, AB = 17 cm and AC = 25 cm. Calculate BC.

In the figure given below, ABCD is a square of side 7 cm. IfAE = FC = CG = HA = 3 cm,(i) prove that EFGH is a rectangle.(ii) find the area and perimeter of EFGH.

In the adjoining figure, D and E are mid-points of the sides BC and CA respectively of a △ABC, right angled at C. Prove that :(i) 4AD2 = 4AC2 + BC2(ii) 4BE2 = 4BC2 + AC2(iii) 4(AD2 + BE2) = 5AB2

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

AD is perpendicular to the side BC of an equilateral △ABC. Prove that 4AD² = 3AB².

In the figure given below, ABCD is a square of side 7 cm. If
AE = FC = CG = HA = 3 cm,
(i) prove that EFGH is a rectangle.
(ii) find the area and perimeter of EFGH.

In the adjoining figure, D and E are mid-points of the sides BC and CA respectively of a △ABC, right angled at C. Prove that :
(i) 4AD² = 4AC² + BC²
(ii) 4BE² = 4BC² + AC²
(iii) 4(AD² + BE²) = 5AB²

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