A cube \(ABCDEFGH\) is pictured below. \(R\), \(S\), and \(T\) are the midpoints of \(A B, F G\) and \(E H\) as shown.
Calculate the size of the angle \(TRS\), giving your answer to one decimal place. (4 marks)
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\(S T=10\)
\(\text{Find} \ \ RT \ \ \text{using} \ \ \triangle RQT:\)
\(R Q=10\)
\(\text{In} \ \ \triangle QET, QE=ET=5\)
\(\text{By Pythagoras }\)
\(QT=\sqrt{5^2+5^2}=\sqrt{50}\)
\(RT=\sqrt{10^2+(\sqrt{50})^2}=\sqrt{150}\)
\(\text{By symmetry,} \ \ RS=\sqrt{150}\)
\(\text{Using cosine rule in} \ \ \triangle RST:\)
\(\cos \angle TRS=\dfrac{(\sqrt{150})^2+(\sqrt{150})^2-10^2}{2 \times \sqrt{150} \times \sqrt{150}}=\dfrac{2}{3}\)
\(\therefore \angle T R S=\cos ^{-1}\left(\dfrac{2}{3}\right)=41.18^{\circ} \ldots=41.2^{\circ}\)
