jumps, remove all seven red checkers, leaving the single black checker. Record your solution.
D C B A 1 2 3 4 D E C B A 1 2 3 4 5
You can observe a lot just by watching. YOGI BERRA
L E S S O N
1.7
A Picture Is Worth a
Thousand Words
A
picture is worth a thousand words! That expression certainly applies to geometry. A drawing of an object often conveys information more quickly than a long written description. People in many occupations use drawings and sketches to communicate ideas. Architects create blueprints. Composers create musical scores. Choreographers visualize and map out sequences of dance steps. Basketball coaches design plays. Interior designers—well, you get the picture.Visualization skills are extremely important in geometry. So far, you have visualized geometric situations in every lesson. To visualize a plane, you pictured a flat surface extending infinitely. In another lesson, you visualized the number of different ways that four lines can intersect. Can you picture what the hands of a clock look like when it is 3:30?
By drawing diagrams, you apply visual thinking to problem solving. Let’s look at some examples that show how to use visual thinking to solve word problems.
Volumes 1 and 2 of a two-volume set of math books sit next to each other on a shelf. They sit in their proper order: Volume 1 on the left and Volume 2 on the right. Each front and back cover is 18inch thick, and the pages portion of each book is 1 inch thick. If a bookworm starts at the first page of Volume 1 and burrows all the way through to the last page of Volume 2, how far will she travel? Take a moment and try to solve the problem in your head.
Bookplate for Albert Ernst Bosman, M. C. Escher, 1946
©2002 Cordon Art B. V.–Baarn–Holland. All rights reserved.
LESSON 1.7 A Picture Is Worth a Thousand Words 73
EXAMPLE A
12 11 10 9 3 2 1 8 7 6 5 4 0 1 5 3 4 6Did you get 214inches? It seems reasonable, doesn’t it?
Guess what? That’s not the answer. Let’s get organized. Reread the problem to identify what information you are given and what you are trying to find. You are given the thickness of
each cover, the thickness of the page portion, and the position of the books on the shelf. You are trying to find how far it is from the first page of Volume 1 to the last page of Volume 2. Draw a picture and locate the position of the pages referred to in the problem.
Now “look” how easy it is to solve the problem. She traveled only 14 inch through the two covers!
In Reasonville, many streets are named after famous mathematicians. Streets that end in an “s” run east–west. All other streets might run either way. Wiles Street runs perpendicular to Germain Street. Fermat Street runs parallel to Germain Street. Which direction does Fermat Street run?
Did you make a diagram? You can start your diagram with the first piece of information. Then you can add to the diagram as new information is added. Wiles Street ends in an “s,” so it runs east–west. You are trying to find the direction of Fermat Street.
Solution
EXAMPLE B
Solution
First page of Volume 1 Last page of Volume 2 MathematicsThe French mathematician Pierre de Fermat (1601–1665) developed analytic geometry. His algebraic approach is what made his influence on geometry so strong.
Sophie Germain (1776–1831), a French mathematician with no formal education, wrote a prize treatise, contributed to many theories, and worked extensively on Fermat’s Last Theorem.
Andrew Wiles (b 1953), an English mathematician at Princeton University, began trying to prove Fermat’s Last Theorem when he was just 10 years old. In 1993, after spending most of his career working on the theorem, sometimes in complete isolation, he announced a proof of the problem.
LESSON 1.7 A Picture Is Worth a Thousand Words 75
Initial diagram Improved diagram Final diagram
The final diagram reveals the answer. Fermat Street runs north–south.
Sometimes there is more than one point or even many points that satisfy a set of conditions. The set of points is called a locus of points. Let’s look at an example showing how to solve a locus problem.
Harold, Dina, and Linda are standing on a flat, dry field reading their treasure map. Harold is standing at one of the features marked on the map, a gnarled tree stump, and Dina is standing atop a large black boulder. The map shows that the treasure is buried 60 meters from the tree stump and 40 meters from the large black boulder. Harold and Dina are standing 80 meters apart. What is the locus of points where the treasure might be buried? Start by drawing a diagram based on the information given in the first two sentences, then add to the diagram as new information is added. Can you visualize all the points that are 60 meters from the tree stump? Mark them on your diagram. They should lie on a circle. The treasure is also 40 meters from the boulder. All the possible points lie in a circle around the boulder. The two possible spots where the treasure might be buried, or the locus of points, are the points where the two circles intersect.
Germain Fermat N S Wiles W E Germain N S Wiles W E Wiles W E H 80 m D 60 m H D 60 m H D 40 m Treasure here Or here
Initial diagram Improved diagram Final diagram