Question 8: how many miles does 1 cm represent?
If the map is accurately drawn and the street from A to K is 6 miles long, how many miles does 1 cm of the map represent?
- Measure A to K on the map with your ruler. Suppose it reads 8 cm. Your printing may measure a little differently; carry your own number through the same steps.
- Those 8 cm stand for 6 miles, so 1 cm stands for 6 ÷ 8 = 0.75 miles.
- That single number now answers question 7. Measure A to F, say 4 cm, and convert: 4 × 0.75 = 3 miles.
- Check yourself: A to F is part of A to K, so the answer must come out under 6 miles. It does.
Question 15: line versus line segment
What is the difference between a line and a line segment?
- Answer from the definitions on page 9, not from a picture. A line segment is part of a line bounded by two endpoints.
- A line has no endpoints; it extends without end in both directions.
- So the difference: a segment has endpoints, and because it does, it has a length you can measure. A line has neither an endpoint nor a length.
- Question 16 leans on this immediately: AB, CD, and HK are three different segments, yet a single line contains them all.
Questions 1 and 2: the first triple of lines
Draw the lines AI, CF, and DG, extending each across the whole figure. What seems to be true about them?
- Copy the figure at least twice the printed size. The effect hides in a small drawing.
- Draw AI, CF, and DG in one color, and extend each one until it leaves the figure. A short stroke can miss the crossing.
- Look at where they cross. All three pass through a single point.
- Page 9 gave the word for this: the three lines are concurrent.
- Expect the same from the triples in questions 3 and 4, and again after you distort the figure in question 5. That persistence is the discovery.
Questions 1 to 4: ranking by eye
Without doing any measuring, which side is the longest? Which angle is the largest? Which side is the shortest? Which angle is the smallest?
- Rank the sides by eye alone, comparing them in pairs: pick the longest against each of the others, then settle the remaining two.
- Rank the angles the same way: widest opening first, narrowest last.
- Write both rankings down before touching a ruler or protractor. Questions 5 to 12 have you measure everything, and the measurements will grade your eye.
- When the two rankings sit side by side, compare them. The largest angle sits opposite the longest side, and the smallest angle opposite the shortest side. That pairing holds for every triangle, and the book proves it in chapter 5.
Question 13: the belt of Orion
What word describes the apparent relation of the points labeled X, Y, and Z in Orion's belt?
- The question wants one vocabulary word, so run through the page 9 list: collinear, coplanar, concurrent.
- X, Y, and Z are points, which rules out concurrent; that word describes lines.
- The three belt stars appear to lie on one straight line, and points that lie on one line are collinear.
- Note the word "apparent" in the question. Seen from Earth the stars line up; in space they sit at different distances along your line of sight.
Set III: the distance around Mars
When the sun is directly overhead at Botrodus, it is 24° from the vertical at Aquae Calidae. The cities are 880 miles apart. What is the distance around Mars?
- The 24° measured at Aquae Calidae equals the angle at the center of Mars between the two cities. That is the same equality Euclid supplied for the earth on page 15.
- Ask how many of those slices fill a full circle: 360 ÷ 24 = 15.
- So the trip around Mars is 15 of the city-to-city distances: 15 × 880 = 13,200 miles.
- The real circumference of Mars is about 13,300 miles, so the book's invented Martian survey lands close to the truth.
Questions 6 to 9: one square, measured twice
A square has sides 1 foot long. What is its area in square feet, and its perimeter in feet? Now think of the same square as having sides 12 inches long. What is its area in square inches, and its perimeter in inches?
- Question 6: sides of 1 foot, so area = 1 × 1 = 1 square foot.
- Question 7: perimeter = 4 × 1 = 4 feet.
- Question 8: the same sides measure 12 inches, so area = 12 × 12 = 144 square inches.
- Question 9: perimeter = 4 × 12 = 48 inches.
- Now reconcile the two descriptions. The perimeter converted by a factor of 12, since 48 inches is 4 feet. The area converted by 144, which is 12 × 12, because the length and the width each picked up a factor of 12. One square foot is 144 square inches, not 12.
Questions 25 and 26: edges and corners
The line segments in which the faces of the pyramid meet are called its edges. How many edges does the pyramid have? How many corners does it have?
- An edge is where two faces meet, so count meetings rather than lines in a drawing.
- The square base meets each of the four triangular faces in one segment: 4 edges around the bottom.
- Each triangle also meets its two neighbors along a slanted segment running up to the tip: 4 more edges.
- Total: 4 + 4 = 8 edges.
- Corners: the 4 corners of the base plus the tip, so 5. With your model in hand, run a finger along each edge as you count.
Questions 8 and 9: the area of the triangle
How does the area of the triangle compare with that of the original square? Why? What is the area of the triangle?
- The square measures 14 cm on each side, so its area is 14 × 14 = 196 square centimeters.
- The triangle is built from the same four pieces. Cutting and rearranging neither adds paper nor removes it.
- So the areas are equal, and the triangle's area is 196 square centimeters. You never needed a triangle formula.
- Question 10 is the contrast: the boundary does change when the pieces turn, so the triangle's perimeter is not the square's perimeter. Measure both and compare.