Lesson 2

Slicing Solids

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Problem 1

Select all figures for which there exists a direction such that all cross sections taken at that direction are congruent.

A:

triangular pyramid

B:

square pyramid

C:

rectangular prism

D:

cube

E:

cone

F:

cylinder

G:

sphere

Solution

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Problem 2

Imagine an upright cone with its base resting on your horizontal desk. Sketch the cross section formed by intersecting each plane with the cone.

  1. vertical plane not passing through the cone’s topmost point
  2. horizontal plane
  3. diagonal plane

Solution

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Problem 3

Name 2 figures for which a circle can be a cross section.

Solution

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Problem 4

Sketch the solid of rotation formed by rotating the given two-dimensional figure using the dashed vertical line as an axis of rotation.

Dotted vertical line. Next to it is a wavy line, vaguely resembling the number 3.

Solution

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(From Unit 5, Lesson 1.)

Problem 5

Draw a two-dimensional figure that could be rotated using a vertical axis of rotation to give the cone shown.

A cone with a dotted vertical line down its middle.

Solution

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(From Unit 5, Lesson 1.)

Problem 6

A regular hexagon and a regular octagon are both inscribed in the same circle. Which of these statements is true?

A:

The perimeter of the hexagon is less than the perimeter of the octagon, and each perimeter is less than the circumference of the circle.

B:

The perimeter of the octagon is less than the perimeter of the hexagon, and each perimeter is less than the circumference of the circle.

C:

The perimeter of the hexagon is greater than the perimeter of the octagon, and each perimeter is greater than the circumference of the circle.

D:

The perimeter of the octagon is greater than the perimeter of the hexagon, and each perimeter is greater than the circumference of the circle.

Solution

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(From Unit 4, Lesson 11.)

Problem 7

Technology required. Find the perimeter of the figure.

Quadrilateral ABCE. Angles B and C = 90 degrees. Angle E = 70 degrees. Side AB = 8, side BC=10.

Solution

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(From Unit 4, Lesson 10.)

Problem 8

Match each trigonometric function to a ratio. You may use ratios more than once.

Right triangle A B C. Angle A C B is 90 degrees, A C  is x units, B C is y units, A B is z units.
A:

\(\tan(A)\)

B:

\(\tan(B)\)

C:

\(\cos(A)\)

D:

\(\cos(B)\)

E:

\(\sin(A)\)

F:

\(\sin(B)\)

1:

\(\frac{y}{z}\)

2:

\(\frac{x}{z}\)

3:

\(\frac{x}{y}\)

4:

\(\frac{y}{x}\)

Solution

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(From Unit 4, Lesson 6.)

Problem 9

Explain how you know lines \(m\) and \(l\) are parallel.

Triangle A B C. A C on line L. Point D on A B, point E on B C. Line m runs through points D and E. Angles B A C and B D E congruent.

Solution

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(From Unit 1, Lesson 20.)