Deck 7: The Nature of Geometry

Question 1

A plant grows for two months and then adds a new branch. Each new branch grows for two months, and then adds another branch. After the second month, each branch adds a new branch every month. Assume that the growth begins in January. How many branches will there be in March? ​

A) 4
B) 16
C) 2
D) 6
E) 8

Answer 1

2

Question 2

Walking north or south on the earth is defined as walking along the meridians, and walking east or west is defined as walking along parallels. There are points on the surface of the earth from which it is possible to walk a mile south, then a mile east, then a mile north, and be right back where you started. Find one such point. ​

A) ( 66° N 56.9° E )
B) The North Pole
C) ( 66° W 56.9° S )
D) A point on the Equator
E) The South Pole

Answer 2

The North Pole

Question 3

Write the ratio of length to width of a standard brick as a decimal. ​

A) 1.81
B) 1.2
C) 2
D) 1.91
E) 3

Answer 3

2

Question 4

Choose one of the following statements to complete the sentence. Discuss your reasoning for your selection. Euclid's treatment of the fifth axiom...

A) ... shows that it is important to question the nature of the assumptions that are made.
B) ... shows that he was really smart.
C) ... is now known to be wrong.

Question 5

Consider the spheres shown in figure. Do one or both of the lettered curves in the figure at the right seem to be lines?

A) both are lines
B) l is a line (a great circle), but m is not
C) neither is a line

Question 6

Associate the given name with one of the following geometries: Euclidean, hyperbolic, elliptic. ​
Nikolai Lobachevski
​

A) Euclidean
B) hyperbolic
C) elliptic

Question 7

How closely does the ratio of the dimensions of a basketball court (36 ft by 78 ft) approximate τ? ​

A) 4.46
B) 2.11
C) 2.19
D) 2.23
E) 2.17

Question 8

If a window is to be 10 feet wide, how high should it be, to the nearest tenth of a foot, to be a golden rectangle? Assume that the width of the window is larger than height. ​

A) 6.2 feet
B) 8.3 feet
C) 6.9 feet
D) 4.9 feet
E) 12.4 feet

Question 9

True or false?
In Lobachevskian geometry, the sum of the measures of the angles of a triangle is greater than 180 .

Question 10

What is a golden ratio? ​

A) the number
B) the number
C) the number
D) the number
E) the number

Question 11

Write the ratio of length to width of a standard index card as a decimal. ​

A) 1.67
B) 1.74
C) 1.62
D) 1.60
E) 1.58

Question 12

Consider the spheres shown in the figure. The sphere at the left shows two lines. In Euclidean geometry, we know that two lines are either parallel or cross at exactly one point. Discuss this property in relation to what you observe in the figure.

A) Two lines intersect in two points.
B) Two lines are either parallel or cross at exactly two points.
C) Lines are great circles, and m is not a great circle.

Question 13

Name the geometry in which the statement is possible. ​
Was used to measure distances in the construction of the pyramids in ancient Egypt.
​

A) elliptic
B) Euclidean
C) hyperbolic

Question 14

If a canvas for a painting is 8 inches wide, how high should it be, to the nearest inch, to be a rectangle with divine proportions? The canvas is higher than it is wide. ​

A) 26 inches
B) 13 inches
C) 12 inches
D) 11 inches
E) 5 inches

Question 15

Leo Moser studied the effect that two face-to-face panes of glass have on light reflected through the panes. If a ray is not reflected, it has just one path through the glass. If it has one reflection, it can be reflected two ways. For two reflections, it can be reflected three ways. Make a conjecture about the number of paths for n reflections.

A) 1, 2, 3, 5, 7,....
B) 1, 2, 3, 4, 7,....
C) 1, 2, 3, 4, 5,....
D) 1, 2, 3, 5, 8,....
E) 1, 2, 3, 6, 12,....

Question 16

Name the geometry in which the statement is possible. The sum of the measures of the angles of a triangle is greater than 180 .

A) elliptic
B) hyperbolic
C) Euclidean

Question 17

Is the figure Saccheri quadrilateral? ​ ​

A) no
B) yes

Question 18

Find the angle (to the nearest degree). ​

A)
B)
C)
D)
E)

Question 19

On a globe, locate Tokyo, Seattle, and Honolulu. Connect these cities with the shortest paths to form a triangle. Next, use a protractor to measure the angles. What is their sum? ​

A) less than 180°
B) greater than 180°
C) 180°

Question 20

A photograph is to be printed on a rectangle in the divine proportion. If it is 7 cm high, how wide is it, to the nearest centimeter ? ​

A) 23 cm
B) 10 cm
C) 16 cm
D) 13 cm
E) 11 cm

Question 21

Consider the spheres shown in the figure.
The sphere at the left shows two lines. In Euclidean geometry, we know that two lines are either parallel or cross at exactly one point. Discuss this property in relation to what you observe in the figure.

Question 22

A plant grows for two months and then adds a new branch. Each new branch grows for two months, and then adds another branch. After the second month, each branch adds a new branch every month. Assume that the growth begins in January. How many branches will there be in June?
__________ branches

Question 23

Associate the given name with one of the following geometries: Euclidean, hyperbolic, elliptic.
Nikolai Lobachevski

Question 24

If a canvas for a painting is 19 inches wide, how high should it be, to the nearest inch, to be a rectangle with divine proportions? The canvas is higher than it is wide.
​
__________ in

Question 25

On a globe, locate Tokyo, Seattle, and Honolulu. Connect these cities with the shortest paths to form a triangle. Next, use a protractor to measure the angles. What is their sum?
Answer 180 degrees, less than 180 degrees, or greater than 180 degrees.

Question 26

Write the ratio of length to width of a standard brick as a decimal.

Question 27

Name the geometry in which the statement is possible.
The base angles of a Saccheri quadrilateral are right angles.

Question 28

A photograph is to be printed on a rectangle in the divine proportion. If it is 8 cm high, how wide is it, to the nearest centimeter? Assume that the width of the rectangle is larger than the height.
​
__________ cm

Question 29

Find the angle (to the nearest degree).
​
__________°

Question 30

Answer true or false.
​
In Lobachevskian geometry, the sum of the measures of the angles of a triangle is greater than 180°.

Question 31

Name the geometry in which the statement is possible.
Was used to measure distances in the construction of the pyramids in ancient Egypt.

Question 32

If a window is to be 7 feet wide, how high should it be, to the nearest tenth of a foot, to be a golden rectangle? Assume that the width of the window is larger than height.
​
__________ ft

Question 33

Is the figure Saccheri quadrilateral? Answer yes or no.

Question 34

Write the ratio of length to width of a standard index card as a decimal. Express your answer to the nearest hundredth.

Question 35

Walking north or south on the earth is defined as walking along the meridians, and walking east or west is defined as walking along parallels. There are points on the surface of the earth from which it is possible to walk a mile south, then a mile east, then a mile north, and be right back where you started. Find one such point.
Answer The North Pole, The South Pole, or A point on the Equator.

Question 36

What is τ? ​

A) The symbol used to denote the height of rectangle. ​
B) The symbol used to denote the golden ratio of rectangle. ​
C) The symbol used to denote the width of rectangle.

Question 37

Leo Moser studied the effect that two face-to-face panes of glass have on light reflected through the panes. If a ray is not reflected, it has just one path through the glass. If it has one reflection, it can be reflected two ways. For two reflections, it can be reflected three ways. How many possible paths are there for three reflections.
​
__________ paths

Question 38

How closely does the ratio of the dimensions of a polo field (160 yd by 300 yd) approximate τ? Express your answer to the nearest thousandth.
​
Enter the absolute value of the difference between τ and the ratio.

Question 39

Complete the sentence.
Euclid's treatment of the fifth axiom shows that ...

Question 40

Consider the spheres shown in the figure.
Explain what is meant by a great circle on a sphere Answer A circle with diameter equal to that of the sphere or A circle with diameter equal to 1.

Question 41

A building's angle of elevation from a point on the ground 30 m from its base is 47°. Find the height of the building (to the nearest meter).
​
__________ m

Question 42

What is the side opposite ∠B? ​ ​
The side __________.

Question 43

Tell whether it is possible to conclude that the triangles are similar. ​ ​

A) similar
B) not similar

Question 44

What is the width of the largest rectangle with length 24 in. you can cut from a circular piece of cardboard having a radius of 13 in.? (See the figure below.) ​ ​

A) 12 in.
B) 14 in.
C) 6 in.
D) 10 in.
E) 5 in.

Question 45

What is the radius of the largest circle you can cut from a rectangular poster board with measurements 39 in. by 40 in.?
​ ​
__________ in.

Question 46

If the distance from the earth to the sun is million miles, and the angle formed between Venus, the earth, and the sun is (as shown in the figure below), find the distance from the sun to Venus (to the nearest hundred thousand miles).
​ ​
Where ​
__________ hundred thousand miles

Question 47

From a police helicopter flying at 1,440 ft, a stolen car is sighted at an angle of depression of 58 . Find the distance of the car (to the nearest ft) from a point directly below the helicopter. ​

A) 900 ft
B) 1,221 ft
C) 2,304 ft
D) 763 ft
E) 890 ft

Question 48

Find the angle by using either the table or a calculator. Round your answer to four decimal places. ​

A)
B)
C)
D)
E)

Question 49

Find the sine, cosine, and tangent for the angle ​A.
​ ​
Where ​
​ ​

Question 50

Find the angle by using either the table or a calculator.
​

Question 51

Find tan 69° by using either the table or a calculator. Round your answer to four decimal places, if required.
​

Question 52

If the distance from the earth to the sun is million miles, and the angle formed between Venus, the earth, and the sun is (as shown in the figure below), find the distance from the sun to Venus (to the nearest hundred thousand miles). ​ ​

A) 66,300,000 miles
B) 66,500,000 miles
C) 66,400,000 miles
D) 132,500,000 miles
E) 66,100,000 miles

Question 53

What is the radius of the largest circle you can cut from a rectangular poster board with measurements 39 in. by 42 in.? (See the figure below. ) ​ ​

A) 39 in.
B) 19.5 in.
C) 21 in.
D) 18.5 in.
E) 42 in.

Question 54

What is tan B?
​

Question 55

From a police helicopter flying at 1,107 ft, a stolen car is sighted at an angle of depression of 68°. Find the distance of the car (to the nearest ft) from a point directly below the helicopter.
​
__________ ft

Question 56

What is sin B? ​ ​

A)
B)
C)
D)
E)

Question 57

What is the width of the largest rectangle with length 16 in. you can cut from a circular piece of cardboard having a radius of 10 in.?
​ ​
__________ in.

Question 58

Find sin 53° by using either the table or a calculator. Round your answer to four decimal places. ​

A) 0.7984
B) 0.7986
C) 0.8136
D) 0.8009
E) 0.7997

Question 59

A building's angle of elevation from a point on the ground 34 m from its base is 31°. Find the height of the building (to the nearest meter). ​

A) 8 m
B) 20 m
C) 29 m
D) 57 m
E) 18 m

Question 60

What is the side opposite ​ ​

A) b
B) a
C) c

Question 61

Are the triangles similar or not similar?
​

Question 62

If a tree casts a shadow of 11 ft at the same time that a 6-ft person casts a shadow of 2 ft, find the height of the tree (to the nearest foot). ​

A) 33 ft
B) 6 ft
C) 2 ft
D) 17 ft
E) 11 ft

Question 63

The figure contains two similar triangles. Find the unknown measure indicated by a variable. Round your answer to the nearest tenth. ​ ​
Where ​

A)
B)
C)
D) ​
E) ​

Question 64

Given two similar triangles, find b​₁.
​ ​ ​

Question 65

The figure contains two similar triangles. Find the unknown measure indicated by a variable. Round your answer to the nearest tenth.
​ ​
Where ​

Question 66

Given two similar triangles, as shown in the figure, find b​₁. ​ ​ ​

A) ​
B) ​
C) ​
D) ​
E)

Question 67

If a tree casts a shadow of 5 ft 7 in. at the same time that a 5-ft 7-in. person casts a shadow of 1 ft 6 in., find the height of the tree (to the nearest inch).
​
__________ ft __________ in

Question 68

Suppose a 6-ft person wishes to determine the height of a bridge above the bottom of a canyon, as shown in the figure below. To do this, this person stands at one end of the bridge (point A) and looks down to a point directly below the other end (point B). With the help of a companion, point P is determined to form two triangles. Find the distance to the bottom of the canyon, if ft and ft. ​ ​

A) 30 ft
B) 8 ft
C) 5 ft
D) 6 ft
E) 40 ft

Question 69

If a tree casts a shadow of 5 ft 8 in. at the same time that a 6-ft 11-in. person casts a shadow of 4 ft 2 in., find the height of the tree (to the nearest inch). ​

A) 8 ft 4 in.
B) 11 ft 6 in.
C) 9 ft
D) 9 ft 5 in.
E) 9 ft 4 in.

Question 70

List all six angles for the figures given in problem. ​ ​ ​

A) ​
B) ​
C) ​
D) ​
E) ​

Question 71

List the lengths of all six sides for the figures given in the problem. ​ ​
Where ​

A) ​
B) ​
C) ​
D) ​
E) none of these choices

Question 72

Use similar triangles and a proportion to find the height of the house shown in the figure.
​ ​
Where ​
__________ ft

Question 73

Find the unknown angles for the figures.
​ ​
Where ​
​ ​

Question 74

List the lengths of the unknown sides for the figures.
​ ​
Where ​ ​
​

Question 75

Use similar triangles and a proportion to find the height of the house shown in the figure below. ​ ​
Where ​

A) 16 ft
B) 20 ft
C) 18 ft
D) 72 ft
E) 4 ft

Question 76

Given is perpendicular to , and is an equilateral triangle. Is similar to ? ​ ​

A) Yes
B) No

Question 77

Suppose a 6-ft person wishes to determine the height of a bridge above the bottom of a canyon, as shown in the figure below. To do this, this person stands at one end of the bridge (point A) and looks down to a point directly below the other end (point B). With the help of a companion, point P is determined to form two triangles. Find the distance to the bottom of the canyon, if ft and ft.
​ ​
__________ ft

Question 78

If a tree casts a shadow of 11 ft at the same time that a 6-ft person casts a shadow of 1 ft, find the height of the tree (to the nearest foot).
​
__________ ft

Question 79

Suppose a 6-ft person wishes to determine the height of a footbridge connecting two buildings, as shown in the figure below. To do this, this person stands at one end of the bridge (point S) and looks down to a point directly below the other end (point T). With the help of a companion, point P is determined to form two triangles. Find the height of the footbridge, if ft and ft. ​ ​

A) 9 ft
B) 6 ft
C) 72 ft
D) 8 ft
E) 48 ft

Question 80

Suppose a 5-ft person wishes to determine the height of a footbridge connecting two buildings, as shown in the figure below. To do this, this person stands at one end of the bridge (point S) and looks down to a point directly below the other end (point T). With the help of a companion, point P is determined to form two triangles. Find the height of the footbridge, if ft and ft.
​ ​
__________ ft