Download Mathematics in Context Sample Review Questions

Mathematics in Context
Sample Review Questions
1. Write 7.528282828282828. . . as a quotient of two whole numbers. (5)
2. Reproduce the proof that
√
2 is irrational. (8)
H
3. To prove the Pythagorean theorem (Proposition I.47) for a
right triangle BAC (with right angle at vertex A) Euclid began
by constructing squares on each side of that triangle. One of
the key subsqeuqnt steps was to prove that triangles BCK and
AEC, pictured to the right, are congruent. Why are they?
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K
A
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B
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D
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C
E
Mathematics in Context, Exam 1, 2/7/2011
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4. This question is a conceptual one relating to axioms and postulates.
a. Explain the significance of the meaning of the word axiom.
b. Explain how Euclid’s Axiom 5 and Proposition I.29 relate to the circumference of the Earth.
5. Euclid’s Proposition I.15 states, “If two straight lines cut one another, they make the vertical angles
equal to one another.” Below is a paraphrased proof of this proposition.
A
D
E
C
B
i. Begin with straight lines AB and CD that intersect at point E.
ii. Segment AE stands on the straight line CD, so by Prop. I.13 ∠CEA + ∠AED = 2 rt. ∠’s.
iii. Likewise, segment DE stands on the straight line AB, so ∠AED + ∠DEB = 2 rt. ∠’s.
iv. Therefore, ∠CEA + ∠AED = ∠AED + ∠DEB.
v. Therefore, ∠CEA = ∠DEB.
vi. Similarly, we can show that ∠AED = ∠CEB, QED.
a. What is the justification for statement iv? (3)
b. What is the justification for statement v? (3)
c. Verify the claim made in vi. In other words, show that, indeed, ∠AED = ∠CEB. (8)
Mathematics in Context, Exam 1, 2/7/2011
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6. Many attribute the following proof of the Pythagorean theorem to Pythagoras himself.
a
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b
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c
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Begin with a right triangle with hypotenuse c, and sides a and b (refer to the dark triangles above).
From this triangle construct two squares with sides of length a + b, also shown above. The two
squares have the same lengths for their sides, so their areas must be equal.
a. Calculate the area of the left-hand square by adding up the areas of the triangles and squares
that compose it. (4)
b. Prove that the inner figure inside the right-hand square is itself a square. Obviously, all its sides
are equal because each side is of length c. Demonstrate that all its angles are right angles by
referring to some well-known propositions in Euclid’s Elements (you do not have to quote the
propositions verbatim, just explain what the main ideas are). You may want to invent labels for
some of the angles to help clarify your explanation. (5)
c. Calculate the area of the right-hand square by adding up the areas of the triangles and square
that compose it. (4)
d. Finish the proof of the Pythagorean theorem by equating the areas found in parts a and c. Show
the algebraic details. (4)