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Equilateral Triangle Length of AD

XAT 2010 · Quantitative Ability & Data Interpretation · Medium · Quantitative Ability

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In an equilateral triangle ABC, whose length of each side is 3 cm, D is the point on BC such that BD = ½ CD. What is the length of AD?

  1. A.

    5\sqrt{5} cm

  2. B.

    6\sqrt{6} cm

  3. C.

    7\sqrt{7} cm

  4. D.

    8\sqrt{8} cm

  5. E.

    None of the above

Answer

C

Explanation

BD+CD=3  ⟹  BD+2BD=3  ⟹  BD=1 cmBD + CD = 3 \implies BD + 2BD = 3 \implies BD = 1\text{ cm}. Let E be the midpoint of BC (foot of altitude AE). BE=1.5 cm  ⟹  DE=BE−BD=1.5−1=0.5 cmBE = 1.5\text{ cm} \implies DE = BE - BD = 1.5 - 1 = 0.5\text{ cm}. Altitude AE=32×3=332 cmAE = \frac{\sqrt{3}}{2} \times 3 = \frac{3\sqrt{3}}{2}\text{ cm}. In right △ADE\triangle ADE, AD2=AE2+DE2=274+14=284=7  ⟹  AD=7 cmAD^2 = AE^2 + DE^2 = \frac{27}{4} + \frac{1}{4} = \frac{28}{4} = 7 \implies AD = \sqrt{7}\text{ cm}.

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