Worksheet: Parallel and Perpendicular Planes

In this worksheet, we will practice using the properties of parallel and perpendicular planes to solve problems.

Q1:

𝑋 and π‘Œ are two parallel planes, where 𝐴 is a point between the two planes. Two straight lines are drawn from the point 𝐴 such that one intersects the planes 𝑋 and π‘Œ at points 𝐡 and 𝐢 respectively, and the other intersects the planes 𝑋 and π‘Œ at points 𝐷 and 𝐻 respectively. If 𝐴 𝐡 𝐴 𝐢 = 1 3 and the surface area of β–³ 𝐴 𝐻 𝐢 = 4 5 0 c m  , find the surface area of β–³ 𝐴 𝐡 𝐷 .

  • A 300 cm2
  • B 150 cm2
  • C 1β€Žβ€‰β€Ž350 cm2
  • D 50 cm2

Q2:

𝑋 and π‘Œ are two parallel planes, and  𝑀 𝐴 ,  𝑀 𝐡 , and  𝑀 𝐢 are drawn to intersect the plane 𝑋 at the points 𝐴 , 𝐡 , and 𝐢 , respectively, and the plane π‘Œ at the points 𝐷 , 𝐸 , and 𝐹 , respectively, where the point 𝑀 doesn’t belong to any of the planes. Given that 𝑀 𝐴 ∢ 𝑀 𝐷 = 2 ∢ 7 , 𝐴 𝐡 = 1 8 c m , 𝐸 𝐹 = 6 8 c m , and π‘š ∠ 𝐴 𝐡 𝐢 = 9 0 ∘ , determine the area of the β–³ 𝐷 𝐸 𝐹 .

Q3:

Given that the plane 𝐾 𝑧 + 2 π‘₯ + 3 𝑦 = βˆ’ 4 is parallel to the plane 𝐿 𝑦 βˆ’ 2 π‘₯ βˆ’ 2 𝑧 = 3 , find the values of 𝐾 and 𝐿 .

  • A 𝐾 = βˆ’ 2 , 𝐿 = 3
  • B 𝐾 = βˆ’ 3 , 𝐿 = 2
  • C 𝐾 = 2 , 𝐿 = βˆ’ 2
  • D 𝐾 = 2 , 𝐿 = βˆ’ 3

Q4:

𝑋 , π‘Œ , and 𝑍 are three parallel planes intersected by two coplanar straight lines 𝐿  and 𝐿  , where 𝐷 𝐻 𝐻 𝑂 = 4 7 . If 𝐴 𝐢 = 4 4 c m , find the length of 𝐴 𝐡 .

Q5:

, , and are three parallel planes intersected by two coplanar straight lines and such that . If , find the length of .

Q6:

Given that the plane 3 π‘₯ βˆ’ 3 𝑦 βˆ’ 3 𝑧 = 1 is perpendicular to the plane π‘Ž π‘₯ βˆ’ 2 𝑦 βˆ’ 𝑧 = 4 , find the value of π‘Ž .

Q7:

Two 3D shapes lie between two parallel planes. Any other plane which is parallel to the two planes intersects both shapes in regions of the same area. What can you deduce about the shapes?

  • AThey have the same surface area.
  • BThey are both prisms.
  • CThey are congruent.
  • DThey have the same volume.
  • EThey are similar.

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