Worksheet: The Pythagorean Theorem in 3D

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In this worksheet, we will practice using the Pythagorean theorem to solve problems in three dimensions.

Q1:

𝐴 𝐡 𝐢 𝐷 𝐴 β€² 𝐡 β€² 𝐢 β€² 𝐷 β€² is a cube. Determine the lengths 𝐴′𝐡 and 𝐴𝐢.

  • A 𝐴 β€² 𝐡 = 9 7 , 𝐴 𝐢 = 9 7 √ 2
  • B 𝐴 β€² 𝐡 = 9 7 √ 2 , 𝐴 𝐢 = 9 7
  • C 𝐴 β€² 𝐡 = 9 7 , 𝐴 𝐢 = 9 7
  • D 𝐴 β€² 𝐡 = 9 7 √ 2 , 𝐴 𝐢 = 9 7 √ 2

Q2:

𝐴 𝐡 𝐢 𝐷 𝐴 β€² 𝐡 β€² 𝐢 β€² 𝐷 β€² is a cube. Determine the lengths 𝐴′𝐡 and 𝐴𝐢.

  • A 𝐴 β€² 𝐡 = 9 3 , 𝐴 𝐢 = 9 3 √ 2
  • B 𝐴 β€² 𝐡 = 9 3 √ 2 , 𝐴 𝐢 = 9 3
  • C 𝐴 β€² 𝐡 = 9 3 , 𝐴 𝐢 = 9 3
  • D 𝐴 β€² 𝐡 = 9 3 √ 2 , 𝐴 𝐢 = 9 3 √ 2

Q3:

𝐴 𝐡 𝐢 𝐷 𝐴 β€² 𝐡 β€² 𝐢 β€² 𝐷 β€² is a cube. Determine the lengths 𝐴′𝐡 and 𝐴𝐢.

  • A 𝐴 β€² 𝐡 = 5 6 , 𝐴 𝐢 = 5 6 √ 2
  • B 𝐴 β€² 𝐡 = 5 6 √ 2 , 𝐴 𝐢 = 5 6
  • C 𝐴 β€² 𝐡 = 5 6 , 𝐴 𝐢 = 5 6
  • D 𝐴 β€² 𝐡 = 5 6 √ 2 , 𝐴 𝐢 = 5 6 √ 2

Q4:

𝐴 𝐡 𝐢 𝐷 𝐴 β€² 𝐡 β€² 𝐢 β€² 𝐷 β€² is a cube. Determine the lengths 𝐴′𝐡 and 𝐴𝐢.

  • A 𝐴 β€² 𝐡 = 5 , 𝐴 𝐢 = 5 √ 2
  • B 𝐴 β€² 𝐡 = 5 √ 2 , 𝐴 𝐢 = 5
  • C 𝐴 β€² 𝐡 = 5 , 𝐴 𝐢 = 5
  • D 𝐴 β€² 𝐡 = 5 √ 2 , 𝐴 𝐢 = 5 √ 2

Q5:

If 𝑀𝐴𝐡𝐢 is a right triangular pyramid; the length of its edge 𝑀𝐴=59cm, and its base 𝐴𝐡𝐢 is right angled at 𝐴, where 𝐡𝐴=105cm and 𝐢𝐴=36cm, find the height of the pyramid rounded to the nearest hundredth.

Q6:

Let 𝐡𝐢𝐷 be an equilateral triangle of side 96, and 𝐡𝐴 a perpendicular to the plane 𝐡𝐢𝐷 of length 48. Determine the length of the perpendicular from 𝐴 to 𝐢𝐷.

  • A107.3
  • B 4 8 √ 3
  • C96
  • D99

Q7:

𝐴 𝐡 𝐢 𝐷 is a triangular pyramid in which π‘šβˆ π΅π΄πΆ=30∘ and π‘šβˆ π΅π΄π·=90∘. Draw π΅π»βŸ‚π΄πΆ. If 𝐡𝐻 is perpendicular to the plane 𝐴𝐢𝐷, 𝐡𝐻=25, and 𝐴𝐷=65, determine the length of 𝐻𝐷.

  • A69.6
  • B41
  • C 2 5 √ 3
  • D 1 0 √ 6 1

Q8:

𝑁 𝐴 , 𝑁 𝐡 , and 𝑁𝐢 are mutually orthogonal. Given 𝑁𝐴=10, 𝑁𝐡=30, and a point 𝐷 on 𝐴𝐡. Find the length 𝐴𝐷 for which 𝐴𝐡 is perpendicular to the plane 𝑁𝐢𝐷.

  • A 3 √ 1 0
  • B √ 1 0
  • C 9 √ 1 0
  • D 1 0 √ 1 0

Q9:

𝑀 𝐴 𝐡 𝐢 is a regular pyramid whose base 𝐴𝐡𝐢 is an equilateral triangle whose side length is 32 cm. If the length of its lateral edge is 88 cm, find the height of the pyramid to the nearest hundredth.

Q10:

A pyramid is on an equilateral triangular base of 21 cm and is 23 cm high. How long, to the nearest hundredth, is the pyramid’s lateral edge?

Q11:

𝐴 𝐡 𝐢 𝐴 β€² 𝐡 β€² 𝐢 β€² is an inclined prism, and 𝐡𝐢𝐢′𝐡′ is a square. Draw π΅π·βŸ‚π΄π΄β€² with 𝐷 on 𝐴𝐴′. Given that 𝐴𝐡=19, 𝐡𝐷=8, and 𝐴𝐢=21, find the length of 𝐷𝐢.

  • A12
  • B20.6
  • C 3 √ 8 2
  • D29.4

Q12:

In the figure, 𝐴𝐡 lies in the plane 𝑋, and 𝐴𝐢 is perpendicular to 𝑋. Given that 𝐴𝐡=6 and 𝐴𝐢=8, find the length of 𝐡𝐢.

Q13:

𝐴 𝐡 𝐢 is a triangle with π‘šβˆ π΅=60∘ and 𝐡𝐢=23. 𝐢𝐷 is drawn perpendicular to the plane of 𝐴𝐡𝐢, and the perpendicular to 𝐴𝐡 from 𝐷 drawn to meet it at 𝐸. If 𝐷𝐸=23, determine the length of 𝐢𝐷 and the angle between 𝐡𝐷 and the plane of 𝐢𝐷𝐸.

  • A19.92, 4053β€²36.22β€²β€²βˆ˜
  • B19.92, 6326β€²5.82β€²β€²βˆ˜
  • C11.5, 6326β€²5.82β€²β€²βˆ˜
  • D11.5, 2633β€²54.18β€²β€²βˆ˜
  • E30.43, 30∘

Q14:

Triangle 𝐴𝐡𝐢 is right angled at 𝐡, and 𝐡𝐷 is orthogonal to the plane 𝐴𝐡𝐢. A perpendicular 𝐷𝐸 is drawn from 𝐸 on 𝐴𝐢. The area of △𝐴𝐢𝐷 is 1,134, 𝐴𝐡=43.2, and 𝐡𝐢=12.6. Let πœƒ be the angle between 𝐷𝐸 and the plane 𝐴𝐡𝐢. Find tanπœƒ to the nearest thousandth.

Q15:

In the figure, suppose that 𝐴𝐡=28, π΅π·βŸ‚ plane 𝐡𝐴𝐢, and π·πΈβŸ‚π΄πΆ. Find the length of 𝐷𝐸.

  • A 7 √ 6 5
  • B √ 2 1
  • C 7 √ 5
  • D25.24

Q16:

In triangle 𝐴𝐡𝐢, π‘šβˆ π΄=60∘ and 𝐴𝐡=24. 𝐡𝐷 is a normal to the plane 𝐴𝐡𝐢, and 𝑂 the foot of the perpendicular from 𝐷 to 𝐴𝐢. If 𝐷𝑂=72, determine the length of 𝐡𝐷.

  • A 1 2 √ 3 9
  • B 4 8 √ 2
  • C 1 2 √ 3 5
  • D 1 2 √ 3 3

Q17:

Rectangle 𝐴𝐡𝐢𝐷 has 𝐴𝐡=25 and 𝐡𝐢=36. Suppose perpendiculars 𝐡𝐻 and 𝐴𝑂 both of length 27. What is the area of 𝐢𝐷𝑂𝐻?

  • A900
  • B675
  • C1,125
  • D1,095.7

Q18:

Determine the surface area of the shown rectangular prism, rounding the result to the nearest tenth.

Q19:

A piece of paper in the shape of a circular sector having a radius of 72 cm and an angle of 275∘ is folded in a way so that the points 𝐴 and 𝐡 meet to form a circular cone of the greatest possible area. Determine the cone’s height to the nearest hundredth.

Q20:

Given that 𝐴𝐡𝐢𝐷𝐸𝐹𝐺𝐻 is a cube whose edge length is 6√2 cm, and 𝑋 is the midpoint of 𝐴𝐡, find the area of the plane π·π‘‹π‘ŒπΈ.

  • A 3 6 √ 5 cm2
  • B72 cm2
  • C101.82 cm2
  • D90 cm2

Q21:

𝐴 𝐡 𝐢 𝐷 𝐴 β€² 𝐡 β€² 𝐢 β€² 𝐷 β€² is a rectangular parallelepiped whose three dimensions are 𝐴𝐡=69cm, 𝐡𝐢=55cm, and 𝐴𝐴′=92cm. Determine the area of the rectangle 𝐢𝐡𝐴′𝐷′.

  • A 6,348 cm2
  • B 6,325 cm2
  • C 8,118 cm2
  • D 3,795 cm2

Q22:

Identify a pair of points between which a diagonal of the cuboid can be drawn.

  • A 𝐹 , 𝐢
  • B 𝐺 , 𝐷
  • C 𝐴 , 𝐢
  • D 𝐴 , 𝐺
  • E 𝐸 , 𝐷

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