Q3 of 42 Page 220

Prove that three times the square of any side of an equilateral triangle is equal to four times the square of the altitude.


Given, an equilateral triangle ABC, in which AD perpendicular BC


Need to prove that 3 AB2 = 4AD2


⇒ Let AB = BC = CA = a


⇒ In Δ ABD and Δ ACD


⇒ AB = AC, AD = AD and ∠ ADB = ∠ ADC


∴ Δ ABD ≅ Δ ACD


∴ BD = CD =


⇒ Now, in Δ ABD, ∠ D = 90°


∴ AB2 = BD2 + AD2


⇒ AB2 = + AD2


= + AD2


3AB2 = 4 AD2


More from this chapter

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1

Prove that the sum of the squares of the sides of a rhombus is equal to the sum of the squares of its diagonals.

 2

ABC is a right triangle right angled at B. Let D and E be any points on AB and BC respectively.

Prove that AE2 + CD2 = AC2 + DE2 .


 4

PQR is a triangle right angled at P and M is a point on QR such that PM ⊥ QR. Show that PM2 = QM.MR.

 5

ABD is a triangle right angled at A and AC ⊥ BD

Show that


i. AB2 = BC . BD


ii. AC2 = BC . DC


iii. AD2 = BD . CD