ABC is an equilateral triangle Find the ratio of the area of the equilateral triangle described on a side - Maths - Triangles

Question

ABC is an equilateral triangle Find the ratio of the area of the
equilateral triangle
described on a side of the triangle to the area of the equilateral
triangle described on one ofits altitude

Manbar Singh · Accepted Answer

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Let ABC be an equilateral ∆ in which AD⊥BC Let AB = BC =
CA = a unitsIn ∆ADB and ∆ADC,∠ADB = ∠ADC
90° each AB = CA each equal to a AD = AD Common∆ADB
≅ ∆ADC RHS⇒BD = DC cpct⇒BD = DC = a2In
∆ADC, AC2 = AD2 + DC2 Pythagoras Theorem⇒AD2 = AC2 -
DC2⇒AD2 = a2 - a24 = 3a24⇒AD = 3a2Let ADE be the
equilateral ∆ described on the altitude AD of ∆ABC So,
AD = DE = AE = 3a2Let ABF be the equilateral ∆ described on
the side AB of ∆ABC So, AB = BF = AF = aarea of ∆ADE =
34AD2 =34×3a24 = 3316a2area of ∆ABF = 34AB2 =34a2area
of ∆ABFarea of ∆ADE = 34a23316a2 = 43

ABC is an equilateral triangle. Find the ratio of the area of the equilateral triangledescribed on a side of the triangle to the area of the equilateral triangle described on one ofits altitude.

ABC is an equilateral triangle. Find the ratio of the area of the equilateral triangle
described on a side of the triangle to the area of the equilateral triangle described on one ofits altitude.