Geometry
Chapter 8 Answers
Circles and Spheres
Circles occur everywhere in our daily lives. For many people, bicycles are their mode of transportation to and from work or school. Our knowledge of circles and their measurements allows us to easily calculate various measurements on bicycles.
Part B, Circles and Tangent Lines
1. Go to the Bicycle Web site to see a diagram of a bicycle. The distance from a hub of a wheel to the point of contact between the tire and the ground is approximately the radius of the wheel. The distance between the points of contact between the tires and the ground is called the "wheelbase."
a. Look at the sketch of a bicycle frame with wheels below. What are
, the angles made by the ground and the segments connecting the hub of each wheel to the point of contact with the ground? How do you know?
[Each of the measures is 90° because a tangent to a circle is perpendicular to a radius drawn to the point of tangency.]b. Go to the Types of bicycles Web site and scroll down to see various specifications for particular types of bicycles. The first dimension given in "Tire Size" is the diameter of the tire in inches. What is the approximate length of a spoke on each of the tires?
[13"]c. What is the distance between the hubs on this bicycle?
[44"]
2. Go to the Riley Web site to see a picture of a recumbent bicycle.
a. Go to the specifications for this bicycle. On this bicycle, the rear wheel is the same size as on the bicycle above. However, the front wheel is smaller. What is the radius of the front wheel of this recumbent bicycle? What is the wheelbase of this bicycle?
[radius of front wheel = 10"; wheelbase = 42"]b. What is the name of the shape formed by quadrilateral BIKE on this recumbent bicycle? What is the distance between the hubs of this bicycle to the nearest tenth of an inch?
[Quadrilateral BIKE is a trapezoid. The distance between its hubs is 42.1"]
Part C, The Circumference of a Circle
3. Learn how to find the circumference of bicycle wheels.
a. Find the circumference of the wheels on the bicycle referred to in 1b to the nearest tenth of an inch.
[81.7"]b. Find the circumferences of the wheels on the recumbent bicycle referred to above to the nearest tenth of an inch.
[front wheel circumference = 62.8"; rear wheel circumference = 81.7"]
Part D, The Area of a Circle
4. Some bicycle racers now use disc wheels because they are more aerodynamic
than traditional wheels with spokes. Rather than having spokes, the interior
of a disc wheel is a solid disc. Go to the Zipp 900 & 840 Discs
Web site and scroll down to the section titled "840 / 900 Disc Wheels"
to see a picture of such a wheel. For these wheels, it is important to know
the area of the wheel in order to create the interior disc. Find the area
of the 26" disc wheel to the nearest tenth of a square inch.
[530.9 square inches]
Being able to figure out arc measures and arc lengths of arcs is critical to mapping the planets. Below, you will view maps of Venus and Pluto and use given statistics to determine the extent of coverage of each of the maps.
Part A, Arcs and Central Angles
1. Go to the Venus Web site to see general information about Venus.
a. Now, go to the section on this Web site titled "Venusian Map." Click on the map to get a more detailed, larger picture. This map does not cover the whole Venusian surface. It only covers from -66.5 to 66.5 degrees latitude. See the illustration below of how much of the surface this map covers.
b. What is
[133°]c. What is the measure of arc AB?
[133°]
2. Now, go to the Pluto Web site to see general information about Pluto.
a. Go to the section on this Web site titled "Map of the Surface of Pluto." Click on the map to get a more detailed, larger picture. Once again, this map does not cover Pluto's whole surface. Estimate the range in latitude this map covers and sketch a diagram of this range similar to the one given in 1a.
Answer:
b. What is
in your diagram of Pluto?
[165°]c. What is the measure of arc AB in your diagram of Pluto?
[same as answer to 2b, 165°]
Part B, Arc Length and Sectors
3. Go back to the Venus Web site.
a. What is the radius of Venus in kilometers?
[6051.8 km]b. What is the circumference of Venus to the nearest kilometer?
[38025 km]
4. Go back to the Pluto Web site.
a.What is the radius of Pluto in kilometers?
[1160 km]b. What is the circumference of Pluto to the nearest kilometer?
[7288 km]
5. The map of Pluto covers a larger portion of the planet, yet it covers
a smaller vertical distance. Why is this?
[The circumference of Pluto is much smaller than
the circumference of Venus. So, even though the map of Pluto covers a greater
portion of the planet, it covers a shorter vertical distance.]
Circles and angles can be found in product logos. Below, you will examine some logos and the angles and arcs in them.
Part A, Inscribed Angles
1. Go to the Jet Express Web site to see a logo which uses a circle, an inscribed angle and one other angle.
a. Use a protractor to measure the inscribed angle.
[40°]b. What is the measure of its intercepted arc?
[80°]c. What is the measure of a central angle which intercepts the same arc as in 1b?
[80°]d. What is the measure of the other acute angle in this logo?
[about 70°]e. Is this a central angle? How do you know?
[It is not a central angle. If it were, it should measure close to 80°.]
2. Now, go to the Volkswagen
Web site to see a logo which is probably familiar to you. Use a compass
and protractor to draw a logo which is geometrically similar, but larger
than the one pictured. Give the measures of each of the central and inscribed
angles in your drawing. Use these measures to find the measures of each
of the intercepted arcs of the circle. Also include the measures of the
arcs in your drawing.
Approximate answer:
Part B, Angles Formed by Secants and Tangents
3. Go to the Foreign Auto Tech Web site to see a logo consisting of a triangle formed by two secants and a diameter of the smaller circle.
a. Use your protractor to measure the acute angle formed by the two secants. What type of triangle is this?
[60°; equilateral triangle]b. The angle formed by the two secants intersects the endpoints of the diameter of the circle. What is the measure of the smaller intercepted arc?
[60°]