October 7

Assignment 3

Area Exercises | Similarity Exercises | Circle Exercises

Exercises

Area and Applications:

1) p.44, Ex. 3.2, 3.3, 3.4

2) p. 45, either Ex. 3.7 OR Ex. 3.8 OR p.46, Ex 3.11 OR prove the Pythagorean Theorem using any regular polygon.

3) page 46, Do any two of Ex. 3.15
NOTE: The pairs of polygons are of equal area.

4)

5) With the right angles and lengths as marked in the diagram, find:

PB, PC, PD, and PE.

If you continue the pattern of the figure, making <PEF=900 and FE = 1, what would be the length of PF?

What would the length of the next segment be?

What is the pattern?

HINT: Keep your answers in radical form!

6) One research question is: given a square or a cube of dimension d, can it be decomposed into smaller, not necessarily congruent, squares or cubes of dimension d. The question about the squares ( d = 2) has been answered. The images below show the decomposition for a square for n = 4, 7, and 8:

A square can NOT be decomposed into 2 smaller squares ( i.e. for n = 2). What is the largest integer for which a square can NOT be decomposed?

NOTE: Research has not found a complete answer for d > 3.

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Similarity

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1) pp. 54 - 55, Ex. 4.1, 4.4, 4.6

2) Use the Dilation method to construct two similar quadrilaterals. After measuring their ratios and their angles, write a definition of similar polygons.

a) Can the AAA Theorem for similar triangles be generalized to similar polygons? That is, can you construct two non-similar polygons whose corresponding angles are congruent?

b) Can the SSS Theorem for similar triangles be generalized to similar polygons? That is, can you construct two non-similar polygons whose corresponding sides are proportional but whose corresponding angles are not equal?

3) p. 55, 4.8. Do this problem in Sketchpad!!!

HINT - Find the two triangles and their respective areas. Locate the point Q such that the inside triangle has area 1/2 the area of the original triangle. Calculate the different ratios to find the method of construction. Test your method of construction on another triangle

4) Given any right triangle and the altitude to its hypotenuse:

Prove that the altitude is the geometric mean of the segments into which it separates the hypotenuse.

OR

the above can be stated as:

AD/CD = CD/ BD

or

the above can be stated as:

CD2 = AD * BD

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What is the implication about the construction of irrational numbers? That is - which irrational numbers can be constructed using the geometric mean idea?

 

5) Prove the following theorem:

The bisector of an angle of a triangle separates the opposite side into segments whose lengths are proportional to the lengths of adjacent sides.
 

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Circles

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1) pp. 73 - 77, Ex. 5.1, 5.2, 5.9 and 5.16

2) Prove:

Given two concentric circles, every chord of the greater circle which is tangent to the smaller circle is bisected at its point of tangency.

3) One arrangement of three circles having different radii so that each circle is tangent to the other is shown below. Construct at least three other arrangements.

4)

The circles with centers A and C are both tangent to the line at B.

A secant of the larger circle passes through A, is tangent to the smaller circle at E, and intersects the line at F.

If the radii of the circles are 8 and 3 respectively, find BF.

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