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Question
In equilateral Δ ABC, AD ⊥ BC and BC = x cm. Find, in terms of x, the length of AD.
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Solution 1
In equilateral Δ ABC, AD ⊥ BC.
Therefore, BC = x cm.
Area of equilateral ΔABC = `sqrt3/4 xx "side"^2 = 1/2 xx "base" xx "height"`
= `sqrt3/4 xx x^2 = 1/2 xx x xx "AD"`
AD = `sqrt3/2 x`
Solution 2
In △ADC and △ADB,
AD = AD ...(Common)
∠ADB = ∠ADC ...(Each 90°)
AB = AC ....(Given, ABC is an equilateral triangle)
Thus, △ADC ≅ △ADB
BD = DC = `1/2`BC ...(By cpct)
Hence, BD = `1/2`x
In △ADB,
∠D = 90°
△ADB is right angle triangle,
by Pythagoras theorem,
AB2 = BD2 + AD2
`x^2 = (1/2x)^2 + "AD"^2`
`"AD"^2 = x^2 − (x^2)/4`
`"AD"^2 = 3/4x^2`
`"AD" = (sqrt(3))/2x`
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