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Lesson: Relationship between a Straight Line and a Plane

Worksheet • 18 Questions

Q1:

Find the coordinates of the point of intersection of the straight line ⃑ π‘Ÿ = ( 8 , 2 , βˆ’ 5 ) + 𝑑 ( βˆ’ 7 , βˆ’ 9 , 1 3 ) with the plane ( 9 , 4 , βˆ’ 5 ) β‹… ⃑ π‘Ÿ = βˆ’ 5 9 .

  • A ( 1 , βˆ’ 7 , 8 )
  • B ( 2 , βˆ’ 6 , 9 )
  • C ( βˆ’ 1 , 7 , βˆ’ 8 )
  • D ( βˆ’ 6 , βˆ’ 8 , 1 4 )
  • E ( βˆ’ 7 , βˆ’ 9 , 1 3 )

Q2:

Find the coordinates of the point of intersection of the straight line ⃑ π‘Ÿ = ( βˆ’ 3 , βˆ’ 4 , 7 ) + 𝑑 ( βˆ’ 3 , 1 , βˆ’ 9 ) with the plane ( βˆ’ 3 , βˆ’ 5 , 7 ) β‹… ⃑ π‘Ÿ = 1 9 .

  • A ( βˆ’ 6 , βˆ’ 3 , βˆ’ 2 )
  • B ( βˆ’ 5 , βˆ’ 2 , βˆ’ 1 )
  • C ( 6 , 3 , 2 )
  • D ( βˆ’ 2 , 2 , βˆ’ 8 )
  • E ( βˆ’ 3 , 1 , βˆ’ 9 )

Q3:

Find the coordinates of the point of intersection of the straight line ⃑ π‘Ÿ = ( βˆ’ 2 , βˆ’ 9 , 6 ) + 𝑑 ( 8 , 5 , βˆ’ 3 ) with the plane ( 9 , 4 , βˆ’ 3 ) β‹… ⃑ π‘Ÿ = 2 9 .

  • A ( 6 , βˆ’ 4 , 3 )
  • B ( 7 , βˆ’ 3 , 4 )
  • C ( βˆ’ 6 , 4 , βˆ’ 3 )
  • D ( 9 , 6 , βˆ’ 2 )
  • E ( 8 , 5 , βˆ’ 3 )

Q4:

Find the coordinates of the point of intersection of the straight line ⃑ π‘Ÿ = ( βˆ’ 8 , 8 , 4 ) + 𝑑 ( 5 , βˆ’ 3 , βˆ’ 1 ) with the plane ( 8 , 7 , βˆ’ 3 ) β‹… ⃑ π‘Ÿ = 2 4 .

  • A ( 2 , 2 , 2 )
  • B ( βˆ’ 1 , 7 , 5 )
  • C ( βˆ’ 2 , βˆ’ 2 , βˆ’ 2 )
  • D ( 7 , βˆ’ 1 , 1 )
  • E ( 1 0 , βˆ’ 6 , βˆ’ 2 )

Q5:

Straight line βƒ–     βƒ— 𝐴 𝐡 is parallel to plane 𝑋 , and from a point 𝑀 neither on the line nor in the plane are drawn rays  𝑀 𝐴 ,  𝑀 𝐡 meeting 𝑋 in 𝐷 and 𝐻 . If 𝑀 𝐴 𝐴 𝐷 = 2 9 : : , what is the ratio between 𝐴 𝐡 and 𝐷 𝐻 ?

  • A 2 ∢ 1 1
  • B 2 ∢ 9
  • C 1 1 ∢ 2
  • D 9 ∢ 2

Q6:

In the figure, 𝐴 𝐡 lies in the plane 𝑋 , and 𝐴 𝐢 is perpendicular to 𝑋 . Given that 𝐴 𝐡 = 6 and 𝐴 𝐢 = 8 , find the length of 𝐡 𝐢 .

Q7:

𝐴 𝐡 𝐢 is a triangle with π‘š ∠ 𝐡 = 6 0 ∘ and 𝐡 𝐢 = 2 3 . 𝐢 𝐷 is drawn perpendicular to the plane of 𝐴 𝐡 𝐢 , and the perpendicular to 𝐴 𝐡 from 𝐷 drawn to meet it at 𝐸 . If 𝐷 𝐸 = 2 3 , determine the length of 𝐢 𝐷 and the angle between 𝐡 𝐷 and the plane of 𝐢 𝐷 𝐸 .

  • A11.5, 2 6 3 3 β€² 5 4 . 1 8 β€² β€² ∘
  • B11.5, 6 3 2 6 β€² 5 . 8 2 β€² β€² ∘
  • C30.43, 3 0 ∘
  • D19.92, 6 3 2 6 β€² 5 . 8 2 β€² β€² ∘
  • E19.92, 4 0 5 3 β€² 3 6 . 2 2 β€² β€² ∘

Q8:

Which of the following makes the straight line π‘₯ βˆ’ π‘₯ 𝑙 = 𝑦 βˆ’ 𝑦 π‘š = 𝑧 βˆ’ 𝑧 𝑛 1 1 1 lie on the plane π‘Ž π‘₯ + 𝑏 𝑦 + 𝑐 𝑧 + 𝑑 = 0 ?

  • A π‘Ž 𝑙 + 𝑏 π‘š + 𝑐 𝑛 = 0 and π‘Ž π‘₯ + 𝑏 𝑦 + 𝑐 𝑧 = 0 1 1 1
  • B ( 𝑙 , π‘š , 𝑛 ) β‹… ( π‘Ž , 𝑏 , 𝑐 ) β‰  0
  • C π‘Ž π‘₯ + 𝑏 𝑦 + 𝑐 𝑧 = 0 1 1 1
  • D ( 𝑙 , π‘š , 𝑛 ) Γ— ( π‘Ž , 𝑏 , 𝑐 ) = 0
  • E π‘Ž 𝑙 + 𝑏 π‘š + 𝑐 𝑛 = 0

Q9:

Triangle is right angled at , and is orthogonal to the plane . A perpendicular is drawn from on . The area of is 1 134, , and . Let be the angle between and the plane . Find to the nearest thousandth.

Q10:

Find the point of intersection of the line π‘₯ βˆ’ 6 4 = 𝑦 + 3 = 𝑧 with the plane π‘₯ + 3 𝑦 + 2 𝑧 βˆ’ 6 = 0 .

  • A ( 1 0 , βˆ’ 2 , 1 )
  • B ( 1 8 , 0 , 3 )
  • C ( 2 , βˆ’ 4 , βˆ’ 1 )
  • D ( 2 4 , βˆ’ 3 , 0 )
  • E ( βˆ’ 2 , 4 , 1 )

Q11:

Find, to the nearest second, the measure of the smaller angle between the straight line π‘₯ βˆ’ 7 7 = 𝑦 βˆ’ 7 βˆ’ 5 = 𝑧 βˆ’ 4 1 and the plane 6 π‘₯ βˆ’ 8 𝑦 βˆ’ 5 𝑧 βˆ’ 1 7 = 0 .

  • A 5 2 4 0 β€² 4 5 β€² β€² ∘
  • B 3 7 1 9 β€² 1 5 β€² β€² ∘
  • C 1 6 3 2 β€² 5 3 β€² β€² ∘
  • D 1 2 7 1 9 β€² 1 5 β€² β€² ∘

Q12:

The equation of a straight line is 𝐿 ∢ π‘₯ βˆ’ 1 2 = 𝑦 + 9 βˆ’ 7 = 𝑧 + 5 βˆ’ 5 and the equaton of a plane is 𝑃 ∢ 8 π‘₯ βˆ’ 2 8 𝑦 βˆ’ 2 0 𝑧 + 1 9 = 0 . Which of the following describes the relationship between 𝐿 and 𝑃 ?

  • A 𝐿 βŸ‚ 𝑃 and they do not intersect.
  • B 𝐿 β«½ 𝑃
  • CThey meet at a point but are not perpendicular.
  • D 𝐿 βŠ‚ 𝑃

Q13:

The equation of a straight line is 𝐿 ∢ π‘₯ + 2 7 = 𝑦 βˆ’ 7 7 = 𝑧 + 5 8 and the equaton of a plane is 𝑃 ∢ 7 π‘₯ + 𝑦 βˆ’ 7 𝑧 βˆ’ 2 8 = 0 . Which of the following describes the relationship between 𝐿 and 𝑃 ?

  • A 𝐿 βŠ‚ 𝑃 and they do not intersect.
  • B 𝐿 βŸ‚ 𝑃
  • CThey meet at a point but are not perpendicular.
  • D 𝐿 β«½ 𝑃

Q14:

The equation of a straight line is 𝐿 ∢ π‘₯ βˆ’ 1 9 = 𝑦 + 2 βˆ’ 1 = 𝑧 + 8 4 and the equaton of a plane is 𝑃 ∢ βˆ’ 5 π‘₯ βˆ’ 9 𝑦 + 9 𝑧 βˆ’ 4 = 0 . Which of the following describes the relationship between 𝐿 and 𝑃 ?

  • A 𝐿 β«½ 𝑃 and they do not intersect.
  • B 𝐿 βŸ‚ 𝑃
  • CThey meet at a point but are not perpendicular.
  • D 𝐿 βŠ‚ 𝑃

Q15:

The equation of a straight line is 𝐿 ∢ π‘₯ + 5 2 = 𝑦 βˆ’ 3 3 = 𝑧 βˆ’ 9 2 and the equaton of a plane is 𝑃 ∢ 6 π‘₯ + 9 𝑦 + 6 𝑧 βˆ’ 2 6 = 0 . Which of the following describes the relationship between 𝐿 and 𝑃 ?

  • A 𝐿 βŸ‚ 𝑃 and they do not intersect.
  • B 𝐿 β«½ 𝑃
  • CThey meet at a point but are not perpendicular.
  • D 𝐿 βŠ‚ 𝑃

Q16:

The equation of a straight line is 𝐿 ∢ π‘₯ + 5 2 = 𝑦 βˆ’ 3 5 = 𝑧 + 6 9 and the equaton of a plane is 𝑃 ∢ 2 π‘₯ + 𝑦 βˆ’ 𝑧 + 1 = 0 . Which of the following describes the relationship between 𝐿 and 𝑃 ?

  • A 𝐿 βŠ‚ 𝑃 and they do not intersect.
  • B 𝐿 βŸ‚ 𝑃
  • CThey meet at a point but are not perpendicular.
  • D 𝐿 β«½ 𝑃

Q17:

What is 𝐴 𝐢 ∩ 𝐴 β€² 𝐢 β€² ?

  • A βˆ…
  • B 𝐢 𝐢 β€²
  • C 𝐴 𝐢
  • D 𝐴 𝐴 β€²

Q18:

In which of the following cases is the straight line βƒ–     βƒ— 𝐴 𝐡 parallel to the plane 𝑋 ?

  • A βƒ–     βƒ— 𝐴 𝐡 ∩ 𝑋 = βˆ…
  • B βƒ–     βƒ— 𝐴 𝐡 βŠ‚ 𝑋
  • C βƒ–     βƒ— 𝐴 𝐡 ∩ 𝑋 = { 𝐡 }
  • D βƒ–     βƒ— 𝐴 𝐡 ∩ 𝑋 = { 𝐴 }
  • E βƒ–     βƒ— 𝐴 𝐡 ∩ 𝑋 = βƒ–     βƒ— 𝐴 𝐡
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