Download Math 4600 HW 4 Due Wednesday, October 13 (1) Prove Theorem

Math 4600
HW 4
Due Wednesday, October 13
(1) Prove Theorem 4.2.13: If the diagonals of a quadrilateral bisect each other, then
the quadrilateral is a parallelogram.
(2) Suppose that 4ABC is a right triangle with right angle at C. Let M be the
midpoint of AB. Prove that the area of 4AM C is the same as 4M CB
(3) The length of the radius of the inscribed circle of a regular n-gon is called the
apothem of the n-gon. (You may need to use some trig for parts (b)-(e))
(a) Prove that the area, A of a regualr n-gon can be computed using the formula
aP
where a represents the apothem and P the perimeter of the regular
A=
2
n-gon.
(b) Consider regular n-gons with the following numbers of sides and in which the
radius of the circumscribed circle is 1.
(i) n = 3
(ii) n = 4
(iii) n = 6
(iv) n = 100
(c) What is the length of the apothem of each polygon?
(d) What is the length of each side?
(e) What is the perimeter?
(f) What is the area?
(g) Determine the ratio of the perimeter to the apothem for a regular 100 − gon.
(h) What do you think the ratio Pa would be for a regular 1000 − gon.
(i) What value does Pa approach as n → ∞?
(4) Prove that if two triangles are similar, the lengths or corresponding angle bisectors
are in the same ratio as the lengths of corresponding sides.
(5) Prove that if two chords are equidistant from the center of a circle, then the chords
are congruent.
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