2020 SJI P2 Q8

The figure shows a solid in which HIJKLM and PQRSTU are regular hexagons of sides \(4\text{ cm}\) and \(10\text{ cm}\) respectively. It is given that \(HP=IQ=JR=KS=LT=MU=15\text{ cm}\).

Diagram not drawn to scale.

1. Show that \(\cos\angle MUT=\dfrac15\).[1]
2. Find the exact value of the perpendicular distance from T to MU.[3]
3. Show that the distance between the planes HIJKLM and PQRSTU is \(3\sqrt{21}\text{ cm}\).[4]
4. Find the length of HS.[3]
5. Find \(\angle KHS\) and \(\angle HKS\).[5]