September 23

Assignment 2

Exercises

1) pages 27-28, Ex. 2.1, 2.3, 2.4, 2.5

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2) When triangles are overlapping, people often have trouble seeing the separate or corresponding parts. Try the problem below first without using Sketchpad. Then use Sketchpad. Notice which you prefer as a method to explore a problem: to work with a given diagram or to work with a diagram you have constructed.

Construct the overlapping triangle diagram which has

< D = < DKM

and

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3) Use Sketchpad to find a relationship(s) between Then prove your conjecture(s).

a) Construct a parallelogram and one diagonal.

b) Construct segments from opposite vertices which are perpendicular to the diagonal.

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4) Midpoints of a Quadrilateral

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a) STEPS:

1) Construct a quadrilateral ABCD.

2) Construct the midpoints of the
four sides: E, F, G, and H.

3) Connects these midpoints
to form a new quadrilateral EFGH.

4) Drag the vertices or the sides of the outer quadrilateral. Be sure to drag it into a variety of sizes and shapes.

5) What conjectures can you make about the quadrilateral formed by the midpoints?

b) Choose one of the following possibilities.

Construct the most general quadrilateral whose midpoint quadrilateral is either:
1) a rhombus, or
2) a rectangle, or
3) a parallelogram.

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5) Although you might want to use Sketchpad to create the diagram, use a geometric proof to prove the following:

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6) For the figure at the right and with the hypothesis given, make a list of all the conclusions :

In this figure, the hypothesis is:

 
Do NOT prove your conclusions, but your conclusions ought to be able to be proven!
 

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7) A Puzzle with an Isosceles Trapezoid

1) Construct an equilateral triangle.

2) Construct D and E, midpoints of two sides of the equilateral triangle.

3) Select the four points: A, D, E, B and construct segments joining these points.

4) Hide the sides of the triangle so that you are left with an isosceles trapezoid ADEB.

5) Construct segments and move them as necessary to divide the trapezoid into four congruent parts.

NOTE: A trapezoid can easily be divided into three congruent parts using the top little triangle (see below) - much harder to divide it into four congruent parts.

Prove:

a) Triangle BCD is an equilateral triangle.

b) Quadrilateral ACDE is a square.

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