Worksheet: Equations of Parallel and Perpendicular Planes

In this worksheet, we will practice finding the equation of a plane that is parallel or perpendicular to another plane given its equation or some properties.

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 𝐴𝐡𝐴𝐢=13 and the surface area of △𝐴𝐻𝐢=450cm, find the surface area of △𝐴𝐡𝐷.

Q2:

𝑋 and π‘Œ are two parallel planes, and 𝑀𝐴, 𝑀𝐡, and 𝑀𝐢 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 π‘€π΄βˆΆπ‘€π·=2∢7, 𝐴𝐡=18cm, 𝐸𝐹=68cm, and π‘šβˆ π΄π΅πΆ=90∘, determine the area of the △𝐷𝐸𝐹.

Q3:

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

  • A𝐾=2, 𝐿=βˆ’2
  • B𝐾=2, 𝐿=βˆ’3
  • C𝐾=βˆ’2, 𝐿=3
  • D𝐾=βˆ’3, 𝐿=2

Q4:

𝑋,π‘Œ, and 𝑍 are three parallel planes intersected by two coplanar straight lines 𝐿 and 𝐿, where 𝐷𝐻𝐻𝑂=47. If 𝐴𝐢=44cm, find the length of 𝐴𝐡.

Q5:

𝑋,π‘Œ, and 𝑍 are three parallel planes intersected by two coplanar straight lines 𝐿 and 𝐿 such that 𝐷𝐻𝐻𝑂=13. If 𝐴𝐢=48cm, find the length of 𝐡𝐢.

Q6:

Given that the plane 3π‘₯βˆ’3π‘¦βˆ’3𝑧=1 is perpendicular to the plane π‘Žπ‘₯βˆ’2π‘¦βˆ’π‘§=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 volume.
  • BThey are both prisms.
  • CThey are similar.
  • DThey have the same surface area.
  • EThey are congruent.

Q8:

Determine the general equation of the plane that contains the straight line π‘₯+27=π‘¦βˆ’65=𝑧+95 and that is perpendicular to the plane βˆ’π‘₯+π‘¦βˆ’2𝑧=2.

  • A7π‘₯+5𝑦+5π‘§βˆ’26=0
  • B7π‘₯+5𝑦+5𝑧+29=0
  • C5π‘₯βˆ’3π‘¦βˆ’4π‘§βˆ’8=0
  • D5π‘₯+3π‘¦βˆ’4π‘§βˆ’44=0
  • Eβˆ’2π‘₯+6π‘¦βˆ’9𝑧+12=0

Q9:

Find the general equation of the plane which passes through the point (2,8,1) and is perpendicular to the two planes βˆ’6π‘₯βˆ’4𝑦+6𝑧=βˆ’5 and 5π‘₯+3π‘¦βˆ’6𝑧=3.

  • A3π‘₯+3𝑦+π‘§βˆ’31=0
  • Bβˆ’3π‘₯βˆ’2𝑦+3𝑧+19=0
  • C5π‘₯+3π‘¦βˆ’6π‘§βˆ’28=0
  • D2π‘₯+8𝑦+𝑧+78=0
  • E3π‘₯βˆ’3𝑦+𝑧+17=0

Q10:

Find the equation of the plane passing through the point (π‘Ž,𝑏,𝑐) and parallel to the plane π‘₯+𝑦+𝑧=0.

  • Aπ‘Žπ‘₯+𝑏𝑦+𝑐𝑧=π‘Ž+𝑏+𝑐
  • Bπ‘₯π‘Ž=𝑦𝑏=𝑧𝑐
  • Cπ‘Žπ‘₯+𝑏𝑦+𝑐𝑧=1
  • Dπ‘₯+𝑦+𝑧+π‘Ž+𝑏+𝑐=0
  • Eπ‘₯+𝑦+𝑧=π‘Ž+𝑏+𝑐

Q11:

Find the general form of the equation of the plane that passes through the two points 𝐴(2,5,4) and 𝐡(3,βˆ’3,5) and that is perpendicular to the plane 26π‘₯βˆ’13𝑦+26π‘§βˆ’26=0.

  • Aβˆ’6π‘₯+6π‘§βˆ’12=0
  • B3π‘₯βˆ’3𝑦+5π‘§βˆ’11=0
  • Cπ‘₯βˆ’2𝑦+𝑧+4=0
  • D2π‘₯+5𝑦+4π‘§βˆ’11=0
  • Eπ‘₯βˆ’8𝑦+𝑧+34=0

Q12:

Find the general form of the equation of the plane passing through the point (4,βˆ’1,1) and parallel to the plane 5π‘₯+6π‘¦βˆ’7𝑧=0.

  • A4π‘₯βˆ’π‘¦+𝑧+7=0
  • B9π‘₯+5π‘¦βˆ’6𝑧+7=0
  • C4π‘₯βˆ’π‘¦+𝑧=0
  • D5π‘₯+6π‘¦βˆ’7π‘§βˆ’7=0
  • E5π‘₯+6π‘¦βˆ’7𝑧+7=0

Q13:

Find the general equation of the plane that is perpendicular to the plane βˆ’6π‘₯+3𝑦+4𝑧+4=0 and cuts the π‘₯- and 𝑦-axes at (5,0,0) and (0,1,0) respectively.

  • A4π‘₯+20π‘¦βˆ’9π‘§βˆ’20=0
  • B4π‘₯+3𝑦+4π‘§βˆ’3=0
  • Cβˆ’6π‘₯+3𝑦+4𝑧+30=0
  • Dβˆ’6π‘₯+3𝑦+4π‘§βˆ’3=0
  • E4π‘₯βˆ’20π‘¦βˆ’9𝑧+20=0

Q14:

Determine if the planes ⟨2,3,4βŸ©β‹…=14r and ⟨4,6,8βŸ©β‹…=34r are parallel or perpendicular.

  • AParallel
  • BPerpendicular

Q15:

Determine if the planes ⟨2,4,13βŸ©β‹…=25r and ⟨3,5,βˆ’2βŸ©β‹…=16r are parallel or perpendicular.

  • APerpendicular
  • BParallel

Q16:

Determine whether the planes ⟨4,2,1βŸ©β‹…=8r and βˆ’π‘₯5+3𝑦10+𝑧5=1 are parallel or perpendicular.

  • AParallel
  • BPerpendicular

Q17:

Determine whether the planes ⟨8,4,10βŸ©β‹…=19r and 4π‘₯+2𝑦+5𝑧=20 are parallel or perpendicular.

  • AParallel
  • BPerpendicular

Q18:

Determine whether the planes ⟨2,3,βˆ’2βŸ©β‹…=12r and π‘₯+2𝑦+4𝑧=6 are parallel or perpendicular.

  • APerpendicular
  • BParallel

Q19:

Determine whether the planes π‘₯+3𝑦+4𝑧=6 and π‘₯5+3𝑦5+4𝑧5=1 are parallel or perpendicular.

  • APerpendicular
  • BParallel

Q20:

Determine whether the planes (π‘₯βˆ’1)+4(π‘¦βˆ’2)+3(𝑧+2)=0 and π‘₯5+4𝑦5+3𝑧5=1 are parallel or perpendicular.

  • APerpendicular
  • BParallel

Q21:

Determine whether the planes 12(π‘₯βˆ’2)+6(π‘¦βˆ’4)+9(𝑧+1)=0 and 4π‘₯+2𝑦+3𝑧=7 are parallel or perpendicular.

  • AParallel
  • BPerpendicular

Q22:

Determine whether the planes 4(π‘₯βˆ’1)+(π‘¦βˆ’2)βˆ’7(𝑧+2)=0 and π‘₯+3𝑦+𝑧=5 are parallel or perpendicular.

  • AParallel
  • BPerpendicular

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