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Question

If ΔABC is an equilateral triangle such that AD is perpendicular to BC, then AD2 = ___.


A

32DC2

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B

2DC2

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C

3 CD2

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D

4 DC2

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Solution

The correct option is C

3 CD2


We first note that the triangles, ADC and ADB are congruent by RHS criterion of congruence.
(AB=AC..( ABC is equilateral)
AD=AD ..(Common hypotenuse)
ADB=ADC=90)
Now, using Pythagoras' theorem in ADC as ADBC, we have
AC2=AD2+CD2.
AD2=AC2CD2
AD2=BC2CD2 ( ABC is equilateral)
AD2=(2CD)2CD2 (BD=CD by CPCT)
AD2=3CD2


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