lesson 9-6 spheres and sectionsmissbarker316.weebly.com/.../39757470/smp_segeo_c09_l06.pdf · 2020....

550 Three-Dimensional Figures

Lesson

Spheres and Sections

Chapter 9

9-6

BIG IDEA The intersection of a plane with a sphere is either a single point or a circle.

Vocabulary

sphere

radius of a sphere

center of a sphere

diameter of a sphere

great circle of a sphere

hemisphere

small circle of a sphere

plane section

2-dimensional cross section

conic sections

a. What is the measure of the arc intercepted by the minute and hour hands of a clock at 4 o’clock?

b. What is the measure of an inscribed angle that intercepts the arc described in Part a?

c. At what time is the measure of the arc between the hour and the minute hands 60º?

Mental Math

Ancient Greek astronomers deduced that Earth was shaped like a sphere based on the fact that Earth casts a consistent round shadow on the Moon during lunar eclipses. Their theory was strengthened by the fact that, at sea, people saw only the tops of ships that were far away. Nevertheless, many Europeans in the Middle Ages and Renaissance believed that the world was fl at. That belief persisted until 1522, when the voyage begun by Ferdinand Magellan, a Portuguese navigator, completed the fi rst circumnavigation of Earth.

The Sphere

A sphere is a 3-dimensional counterpart of the circle. Notice that the terminology of circles extends to spheres.

Defi nitions of Sphere, Radius, Center

A sphere is the set of points in space at a certain distance (its radius) from a point (its center).

A sphere is a surface like a table tennis ball or a bubble, not a solid fi gure. A radius of a sphere is any segment connecting the center of the sphere to a point of the sphere. A diameter of a sphere is any segment connecting two points of the sphere that contains the center of the sphere.

To draw a sphere, draw a circle. Then, to give the illusion of depth, draw an oval through the middle using dashed lines for the top half of the oval. You may need to add the center, the radius, the diameter, or arcs of the circle, as in the sphere with center A shown at the right.

D C

B

A

SMP_SEGEO_C09L06_550-558.indd 550SMP_SEGEO_C09L06_550-558.indd 550 5/21/08 12:27:58 PM5/21/08 12:27:58 PM

Spheres and Sections 551

Lesson 9-6

As with a circle, the center and all points inside the sphere are not points of the sphere. If you wish to include the interior of the sphere, then you need to call it a solid sphere (just as you did for prisms and pyramids). To picture a solid sphere, you may wish to shade the drawing, as shown at the right.

QY1

Sections of a Sphere

In 1884, Edwin A. Abbott wrote Flatland, a novel about a 2-dimensional world. The characters in the novel are squares, triangles, other polygons, circles, and line segments. The story hinges on a sphere from a 3-dimensional world passing through Flatland and befriending a square. As the sphere passes through the plane of Flatland, the sphere appears to the square as a point, then a circle getting bigger and bigger, for a while. Then the circle gets smaller and smaller until it becomes a point and disappears. The square is seeing a part of the cross section of the sphere as it intersects the plane.

Two types of intersections can occur with the intersection of a sphere and a plane. The intersection is a single point if the plane just touches the sphere; it is a circle otherwise. (In Question 16, you are asked to prove that the intersection is a circle.) If the plane contains the center of the sphere, the intersection is called a great circle of the sphere. A great circle (shown below at the right) splits the sphere into two hemispheres. Otherwise, the intersection is called a small

circle (below in the center).

OA

point small circle great circle

QY1

Name the center, a diameter, and two radii in the sphere on page 550.

A scene from Flatland: The

Movie, 2007.



Chapter 9

Both great circles and small circles are examples of plane sections of a sphere.

Defi nition of Plane Section

A plane section of a 3-dimensional fi gure is the intersection of that fi gure with a plane.

Plane sections are also known as 2-dimensional cross sections. Some plane sections in this book are shaded for clarity, even when the section is the boundary of the shaded region.

Activity 1

MATERIALS DGS-3D

The goal is to create a model of the sphere passing through

the plane in Flatland.

Step 1 Construct point O on a plane X. Construct a line perpendicular to plane X through O. Construct a point P on that line as shown at the right.

Step 2 Construct __

AB as shown.

Step 3 Hide point O and �

� OP but don’t hide P.

Step 4 Construct a sphere using P as the center and AB as its radius.

Step 5 Your sphere can be moved up and down by dragging point P. Drag point P so that the sphere intersects the plane. Construct the intersection curve of the sphere and the plane as shown in the diagram at the right.

Step 7 Hide the sphere.

Step 8 Recreate the scene from Flatland by dragging point P up and down. As P approaches the plane, you should see a circle of increasing size. What is the largest radius this circle will have and when will this occur?

Earth as a Sphere

Earth is almost a solid sphere. One of its great circles is the equator. Points on the equator are about 6378 km (or 3963 miles) from the center of Earth. However, Earth has been slightly fl attened by its rotation. The North and South Poles are about 6357 km (or 3950 miles) from the center of Earth.



Lesson 9-6

Below are two sketches of Earth. The sketch at the left is of Earth as seen from slightly north of the equator, so the equator is tilted. Notice how the oval representing the circle of the equator is widened to give the illusion of looking at it from above. Then the South Pole cannot be seen. The sketch at the right is as seen from the plane of the equator.

Earth as seen from abovethe equator and North Pole

North Pole

Earth as seen from plane of the equator

≈ 3963miles

≈ 3950miles

North Pole

South Pole

equator

QY2

In about 230 BCE, the Greek mathematician Eratosthenes used the following information to estimate the circumference of Earth. He noticed that at noon on the summer solstice (on June 21st), the Sun was directly overhead at Syene. He know that Alexandria was about 5000 stades due north of Syene. (The stade was a Greek unit of length equalto about 517 feet.) In another year, he calculated that at noon on the summer solstice in Alexandria, the Sun was 7.2º away from overhead.

Because Alexandria is due north of Syene, Eratosthenes concluded that the distance between the cities was a fraction of Earth’s circumference. He assumed that the Sun was far enough away that rays from anywhere on Earth to it would be parallel, as indicated in the diagram at the right.

Example

Use the above fi gure to show how Eratosthenes used this information to

fi nd the circumference of Earth.

Solution Let C be the circumference of Earth. Due to the

Corresponding Angles Postulate, m∠O = ?

Thus the distance from Alexandria to Syene is ?

____

360 · C.

So, ?

____

360 C = 5000 stades.

Solving this equation, C = ? stades

≈ ? feet

≈ ? miles.

QY3

QY2

Find the difference in the circumferences of Earth around the equator and around the poles.

5000 stades7.2°

Syene

O

Alexandria

GUIDED QY3

Today, with accurate readings from space, we know that the circumference of the section created by intersecting a plane through the North and South Poles is about 24,860 miles. How does this compare with Erastothenes’ measurement?



Chapter 9

Plane Sections of Prisms and Cylinders

In Flatland, the square can only infer the sphere’s shape by looking at its cross sections. While Flatland is fi ction, the same idea is very commonly used today with all sorts of shapes. Biologists use plane sections of tissue to study a tissue’s cell structure. These sections are thin enough so that light from a microscope will shine through them. Doctors use a CAT (computerized axial tomography) scanner to take x-rays of plane sections of the human body for diagnostic purposes. Engineers and architects use cross sections to describe the shapes of things they are designing or examining.

Activity 2

MATERIALS DGS-3D or model of a cube (optional)

The fi gure at the right shows a cube in which the vertices and the midpoints

of the edges are labeled. Identify the plane that cuts the cube to produce

the plane sections described below. Recall that planes can be identifi ed

using three noncollinear points. If you fi nd that you are having a hard time

visualizing this, you can create a DGS 3-D cube and try out the planes that

you think might work. A physical model of a cube might also be helpful.

Description of Plane Section Plane

Example: square HDP

1. scalene triangle ?

2. isosceles triangle ?

3. equilateral triangle ?

4. rectangle that is not a square ?

5. isosceles trapezoid that is not a rectangle ?

6. pentagon ?

7. hexagon ?

Activity 2 shows that a variety of shapes are possible for the plane sections of a cube. A variety of plane sections are possible also for a prism or a cylinder, depending on whether the intersecting plane

1. is parallel to the bases,

2. is not parallel to and does not intersect a base or bases, or

3. intersects one or both bases.

FGH

B

E

A DC

QST

M

I

R

PONL K

J

A plane section of the brain

of a sleeping person.



Lesson 9-6

Activity 3

Consider the fi gures below.

plane sections parallel to bases

Step 1 If the intersecting plane is parallel to the bases, make a conjecture about the relationship between the base and the section.

Step 2 For a prism, if the intersecting plane is neither parallel to nor intersecting the bases, make a conjecture about the relationship between the number of sides of the base and the number of sides of the section.

Step 3 For a cylinder, if the intersecting plane is neither parallel to nor intersecting the bases, make a conjecture about the shape of the section.

When a plane intersects a base as well as lateral faces, more sections are possible. Figure I at the right shows a plane intersecting a solid pentagonal region. The section is the triangular region ABC where

___ AB is on the back face,

___ AC is

on the side face, and ___

BC is on the bottom face of the prism. Figure II shows a plane intersecting a cylinder. The section is a rectangle.

Activity 4

Make drawings like the two plane sections of prisms I and II shown at the right

above, but with a prism that has a quadrilateral base.

Plane Sections of Pyramids and Cones

For pyramids and cones, sections parallel to the base have shapes similar to the base, but they are smaller. You can sketch them by drawing segments or arcs parallel to the base.

plane sections intersecting one or both bases

I

A

C

II

B

plane sections not parallel to bases



Chapter 9

Conic Sections

Below at the far left are two right conical surfaces with the same axis, formed by rotating a line intersecting the axis about that axis. The plane sections formed are called the conic

sections. The conic sections describe orbits of planets and paths of balls and rockets. Their focusing properties are used for long-range navigation (LORAN), telescopes, headlights, satellite dishes, fl ashlights, and whispering chambers. You are likely to have graphed parabolas in your study of algebra and you may have graphed hyperbolas, too. You are likely to study all the conic sections in later mathematics courses.

hyperbola(plane intersecting

both cones)

ellipse(plane not⊥ to axis,

intersecting only one cone)

circle(plane ⊥to axis)

axis

parabola(plane ‖to edge)

axis axis

Questions

COVERING THE IDEAS

1. Draw a sphere with center P and a diameter ___AK that is not

horizontal.

2. Fill in the Blank Complete the analogy: Circle is to 2 dimensions as sphere is to ? .

3. Defi ne: plane section.

4. Name all possible types of plane sections of the object. a. a sphere b. an orange c. a straw

5. True or False Every cross section of a prism with an n-gon base is an n-gon.

6. True or False The diameter of the equator is the same as the diameter of the plane section of Earth containing the poles.

7. a. Name the four types of conic sections. b. Name three uses of conic sections.



Lesson 9-6

In 8−11, draw a box with a plane section that satisfi es the condition.

8. parallel to a face 9. not parallel to a face10. intersecting four faces and not containing any vertex of the cube11. intersecting all six faces and not containing any vertex of the cube

12. If Eratosthenes were alive today, he could repeat his calculation of the circumference of Earth using places in the United States. Louisville, Kentucky is 408 miles due north of Montgomery, Alabama. At a given moment, the Sun is 6º lower in the sky in Louisville than it is in Montgomery. From this information, what estimate would Eratosthenes obtain for the circumference of Earth?

APPLYING THE MATHEMATICS

In 13−15, copy the fi gure shown.

a. Sketch a plane section parallel to the bases.

b. Sketch a plane section not parallel to and not intersecting

the base(s).

c. Name the shape of each section.

13.

right pentagonalpyramid

14.

right cone

15.

16. Here is a proof that the intersection of a sphere and a plane not through its center is a circle. Given is sphere O and plane M, intersecting in the curve containing A and X as shown below. Fill in the justifi cations.

Proof Let P be the foot of the perpendicular from point O to plane M. Let A be a fi xed point and X be any other point on the intersection.

a. ___

OP ⊥ __

PA and ___

OP ⊥ __

PX because ? .

b. ___

OP � ___

OP because of the ? .

c. ___

OA � __

OX because ? .

d. �OPX � �OPA by the ? .

e. __

PX � __

PA by the ? .

Thus, any point X on the intersection lies at the same distance from P as A does. So, by the defi nition of circle, the intersection of sphere O and plane M is the circle with center P and radius PA.

right hexagonalprism

right hexagonalprism

O

A PX

M



Chapter 9

17. Consider the fi gure at the right, in which the radius PC of the cross section is equal to the perpendicular distance QP from the center of the cross section to the sphere. Suppose the area of the cross section is 9π square meters. If the radius of the sphere is 5 meters, fi nd the distance from the center of the sphere to the cross section.

REVIEW

18. Here are three views of a building. (Lesson 9-5)

a. How tall in stories is the building? b. How many blocks long is the building

from front to back? c. Where is the tallest part of the building? d. Create an isometric drawing of the building.

19. Mayra cut a paper party hat that was in the shape of a cone along a straight line, and spread it out on the table to get the fi gure at the right. Trace the fi gure, and draw a segment whose length is equal to the slant height of the hat (before Mayra cut it). (Lesson 9-3)

20. Suppose �ABC is a triangle with no lines of symmetry. Explain why all of the angles in the triangle have different measures. (Lesson 6-2)

21. Let A and B be two points in the plane. Describe the set of all points that are rotation images of B about A. (Lesson 3-2)

22. Look at the networks at the right. For each one, let V equal the number of nodes, and E equal the number of arcs. (Lessons 2-7, 1-3)

a. Calculate E - V + 1 for each of the networks. b. Draw a network where E - V + 1 = 4.

EXPLORATION

23. Earth is often described as an oblate spheroid. The oblateness of a planet measures how much a planet bulges at the equator.

a. Look in a reference for a formula for oblateness. b. What is the oblateness of Earth? c. Which major planet has the greatest oblateness? d. Which major planet is most like a sphere?

5

5

x

O

Q

CP

top view

left right

left

right

front back

front view right view

QY ANSWERS

1. The center is A, ___

DC is a diameter,

__ AB and

__ AC are

two radii.

2. 2π · 3963 - 2π · 3950 = 2 · π · 13 = 26π miles.

3. approximately 1.5%


lesson 9-6 spheres and sectionsmissbarker316.weebly.com/.../39757470/smp_segeo_c09_l06.pdf · 2020....

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