Chapter 9
Chapter Objectives
9.1 Concepts
9.3 Perimeter, Area, and Circumference
9.5 Volumes
9.4 Triangles, Similarity and the Pythagorean Theorem
9.1 Concepts of Euclidean Geometry
Definitions:
A point has no magnitude and no size.
A line has no thickness and no width and it extends indefinitely in two directions.
A plane is a flat surface that extends infinitely.
A capital letter usually represents a point. A line may named by two capital letters representing points that lie on the line or by a single letter such as l. A plane may be named by three capital letters representing points that lie in the plane or by a letter of the Greek alphabet such as
An angle is the union of two rays that have a common endpoint. An angle can be named with the letter marking its vertex,
Angles are classified and named with reference to their degree measure.
Measure / NameBetween 0° and 90°
90°
Greater than 90° but less than 180°
180°
In figure A, the sides of NMP have been extended to form another angleRMQ, the pairNMP and RMQ are called vertical angles.
Question: Another pair of vertical angles has been formed at the same time. Write their name.
Answer: ______, and ______.
If the sum of the measures of two acute angles is 90°, the angles are said to be complementary. For example, angles measuring 40° and 50° are complementary angles.
If two angles have a sum of 180°, they are supplementary. The supplement of an angle whose
measure is 40° is an angle whose measure is 140°.
Question: Use the figure A to indentify a pair of supplementary angles.
Answer: ______, and ______.
Let’s Do It!
Find the measure of each marked angle in the given figure
a)
b)
c)
Parallel lines are lines that lie in the same plane and do not intersect. Figure B shows parallel lines m and n. When a line q intersects two parallel lines, q is called a transversal.
In Figure B, the transversal intersecting the parallel lines forms eight angles, indicated by numbers. Angles 1 through 8 in the figure possess some special properties regarding their degree measures, as shown below
Let’s Di It!
Find the measure of each marked angle in the figure, given that lines m and n are parallel.
Bonus Example
The supplement of an angle measures 10° more than three times its complement. Find the measure of the angle.
Solution
Let x= the measure of the angle( in degrees)
Measure of its supplement
Measure of its compliment
The supplement of an angle measures 10° more than three times its complement supplement=10 + 3*complement
So the angle measure is 50 degrees.
Note t that its supplement (130degrees) is 10 more than three times its complement 40degrees.
Bonus : Let’s Di It!
Half the supplement of an angle is 12° less than twice the complement of the angle. Find the measure of the angle.
End of section 9.1. Start your online homework on MyMathLab.
9.3 Perimeter, Area, and Circumference
Definitions:
A polygon is a simple closed curve made up only of straight line segments.
Question: Give two examples of a polygon?
Answer:
Question: Another Important Geometric shape is the circle. What do we call the perimeter of the circle?
Answer:
Question: Identify the Formula to compute the perimeter/circumference and the area of the following geometric shapes:
a. Triangle b. Rectangle
Perimeter= Perimeter=
Area= Area=
c. Square d. Circle with radius r.
Perimeter= Perimeter=
Area= Area=
e. Circle with diameter d
Perimeter=
Area=
Let’s Do it! Find the area of each figure. Use 3.14 as an approximation for
Let’s Do it!
Figure shows the floor plan of a building, made up of various rectangles. If each length given is in meters, how many square meters of carpet would be required to carpet the building?
Let’s Do it!
a. A square has a perimeter 76 inches. What is the length of each side?
b. The length of a rectangular shape is 3cm more than twice the width. The perimeter is 132cm. Find the dimensions of the rectangle.
Mathematics Goes To Hollywood
Cast Away was one of the top films of 2000. It stars Tom Hanks as Chuck Noland, a Federal Express employee who, as the only survivor in a plane crash, is stranded for 4 years alone on a tropical island. Not long after the crash, speaking to his “friend” Wilson, a volleyball that had washed ashore, he used geometry to assess their chances of being found. Sketching a circle and performing an arithmetic calculation on the side of a rock, Chuck realizes their futility.
So, Wilson. We were en route from Memphis for eleven and a half hours at about 475 miles an hour. They think that we are right here.
But we went out of radio contact and flew around that storm for about an hour. So that’s a distance of what, 400 miles? Four hundred miles squared, that’s 160,000, times pi, 3.14, . . .
Chuck’s calculation for the size of the search area is an application of the formula for the area of a
circle, He looks to Wilson, and sighs,
…That’s twice the size of Texas…They may never find us.
What was Chuck’s answer? The land area of Texas is 261,797 square miles. Was he correct?
End of section 9.3. Start your online homework for this section on MyMathLab.
9.5: Volume, and Surface Area
Space figure: take three dimensions of space to represent the figure.
Volume is a measure of capacity of a space figure. Volume is measured in cubic units.
Surface Area is the total area that would be covered if the space figure were “peeled” and the peel laid flat. Surface area is measured in square units.
Let’s Do It! Volume or surface area
A) In order to determine the amount of liquid a spherical tank will hold, would you need to use volume or surface area?
B) In order to determine the amount of leather it would take to manufacture a basket ball, would you need to use volume or surface area?
Volume of Surface Area of a Box
The volume V and surface area S of a box with length l, width w, and height h is given by the formulas
Example:
Volume of Surface Area of a Right Circular Cylinder
The volume V and surface area S of a right circular cylinder with base radius r and height h is given by the formulas
Example:
MORE ORMULAS:
Sphere:
Right Circular Cone
Pyramid
Let’s Do It! Computing volumes and surface areas. Use
A) A coffee can with radius 6.3 cm and height 15.8cm.
Volume=
Surface area=
B) The great Pyramid of Cheops, near Cairo--it base is a square 230m on a side, while the height is 137m.
Volume=
Surface area=
End of section 9.5. Start your online homework for this section on MyMathLab.
9.4 The Geometry of Triangles, Similarity, and Pythagorean Theorem
In this section we will use the symbolism to designate triangles.
Many of the key ideas of geometry depend on similar triangles, pairs of triangles that are exactly the same shape but not necessarily the same size.
Figure below shows three pairs of similar triangles. (Note: The triangles do not need to be oriented in the same fashion in order to be similar.)
Example
In the figure, is similar to Find the Unknown side lengths in
Solution: As mentioned above, similar triangles have corresponding sides in proportion.
Side EF corresponds to side CB. This leads to another proportion
Now, Use the same technique to find the length of side DF.
Let’s Do It!
Find the measures of the unknown parts of the similar triangles STU and ZXY in the figure.
Solution:
Triangle ZXY
angle Z =______, Angle Y=______.
Measure of side XY=______,
Triangle STU
Angle U=______, Angle T=______.
Measure of side ST=______.
Mathematics Goes To Hollywood
The 1984 Grenada Television Ltd. production The Return of Sherlock Holmes: The Musgrave Ritual, stars Jeremy Brett as Sir Arthur Conan Doyle’s legendary sleuth. In solving the mystery, Holmes is faced with the problem of determining where the tip end of the shadow of a tree 64 feet tall would have been on the ground. Because the tree is no longer standing, he uses similar triangles to solve the problem. He finds that a vertical rod of 6 feet casts a shadow of 9 feet, so he determines that at the same time of day the tree of height 64 feet would have cast a shadow of 96 feet.
The Pythagorean Theorem
Example
Find the length a in the right triangle shown in the figure.
Solution:
Let’s Do it!
A ladder of length 10 cubits has its foot 6 cubits from a wall. To what height does the ladder reach?
Mathematics Goes To Hollywood
In the 1939 classic The Wizard of Oz, Ray Bolger’s character, the Scarecrow, wants a brain. When the Wizard grants him his “Th.D.” (Doctor of Thinkology), the Scarecrow replies with a statement that has made mathematics teachers shudder for almost 70 years. His statement is quite impressive and sounds like the Pythagorean Theorem but is totally incorrect.
The scare crow theorem: in an isosceles triangle, the sum of the square roots of two sides is the square root of the third side.
If you watch this scene, also notice that between camera shots preceding his statement, the Scarecrow’s position changes. These are two errors in a movie that has over 100 reported errors
(See www.moviemistakes.com), yet remains one of the most beloved motion pictures in history.
End of section 9.4. Start your online homework for this section on MyMathLab.