Tessellate S.T.E.M.S. (2019) - Mathematics - School - Set 1 - Objective Problem 3

Geometry Level 3

Let $ABCD$ be a parallelogram such that $\angle BAD = 60^{\circ}$ . Let $K$ and $L$ be the midpoints of $BC$ and $CD,$ respectively. Given that $ABKL$ is a cyclic quadrilateral, find $\angle ABD$ .

This problem is a part of Tessellate S.T.E.M.S. (2019)

60^{\circ}

45^{\circ}

75^{\circ}

72^{\circ}

1 solution

Tessellate S.T.E.M.S. Mathematics
Oct 22, 2018

Let $AB, LK$ intersect at $E$ , and construct $BH$ perpendicular to $AB$ . Assign $EB=:a$ so that $AB=2a$ and $EK=KL=:b$ so that $BD=2b$ . Then $EB \cdot EA = EK \cdot EL \Leftrightarrow 3{a^2} = 2{b^2} \Leftrightarrow b = \frac{{a\sqrt 6 }}{2}$ and $BH = a\sqrt 3 \Leftrightarrow BD = 2b = a\sqrt 3 \sqrt 2 = BH\sqrt 2$ , whence $\omega=45^0$ and finally $\boxed{\angle ABD=75^o}$ .

Tessellate S.T.E.M.S. (2019) - Mathematics - School - Set 1 - Objective Problem 3

1 solution

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