Crane Arm
A parallelogram is formed by joining together four equilateral triangles. What is the length of the longest diagonal?
Age
14 to 16
Challenge level
Problem
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The parallelogram PQRS is formed by joining together four equilateral triangles of side 2 units, as shown.
What is the length of the diagonal SQ?
If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.
Student Solutions
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In the diagram, T is the foot of the perpendicular from Q to the extension of SR. All of the angles in the equilateral triangles are 60 °, so ∠QRT is also 60 °. Then ΔQRT is a right-angled triangle, so we can show that the lengths of RT and QT are $1$ and $\sqrt{3}$ respectively.
Applying Pythagoras' Theorem to ΔQST, $SQ^2 = ST^2 + QT^2 = 5^2 + (\sqrt{3})^2 = 28$.
So the length of SQ is $\sqrt{28}=2\sqrt{7}$ units.