Investigation #3 - Post by March 8th

This investigation is focused on the use of technology for mathematics associated with geometry. Geometer's Sketchpad and Geogebra are essential programs for these problems.

1) Properties of a Rectangle

Part 1:

For any given rectangle,

1. Are the diagonals perpendicular bisectors?

2. Do the diagonals bisect the angles?

Use a dynamic geometry software to investigate these questions. If these questions are not true determine if there are specific conditions that make them true and provide mathematical reasoning to support any conclusions. If they are true, provide mathematical reasoning to support your conclusion.

Part 2:

Examine the properties of two other quadrilaterals using a dynamic geometry software. What properties can be proven and which ones define the shape? Provide mathematical reasoning to support your conclusion.

2) Transformations

Using transformations, provide as many possible ways (at least two) to get from the appearance of starting object to the ending object as you can.

Part 1.

The object in which the yellow triangle is enclosed is a square, which makes the yellow triangle a right triangle. Also, the yellow triangle's height is a quarter of the length of one of the side of the square.

Part 2.

Part 3.

Part 4. Make your own transformation that is not shown above. Explain why you chose it and how to solve it.

3) The Golden Rectangle and Spiral.

Use the following instructions to create the golden rectangle and spiral then investigate some of it's properties.

1) Construct square ABCD and midpoint m.

2) Construct a circle centered at M with radius MB and ray AD.

3) Find the intersection (point F) of ray AD and the circle then construct a perpendicular line to ray AD at point F.

4) Construct ray BC and find the intersection of the perpendicular and ray BC (point G).

5) Construct segments to form a golden rectangle (ABGF)

6) Construct a circle centered a point C with radius BC. You can use this circle to make the first part of the golden spiral, arc BD.

7) Continue this process using segment DF as the length of the original square.

Questions and statements to investigate:

Using side lengths, a golden rectangle can be defined in this way:

The ratio of the sum of two of the adjacent sides to the long side is equal to the ratio of the long side to the short side. Decode this statement algebraically as it refers to the golden rectangle above. Is it true of our construction?

The numeral value of the golden ratio is:

and it can be found by calculating the ratio of the long side (segment AF) to the short side (segment AB). Determine if this calculation is true and show how they are equal.

4) The Regular Pentagon and it's properties

Construct a regular pentagon.

Show how this construction is related to the golden ratio and investigate one other property of the figure.

5) The Medial Triangle

The medial triangle is a triangle formed by connecting all of the medians of a given triangle. Construct the medial triangle and investigate its properties. Is it similar to the original triangle? How does the area of the medial triangle compare to the area of the original triangle? Provide mathematical reasoning to support your answers.

6) Pedal Triangle

The pedal triangle is the triangle formed by the intersections of the perpendiculars from a given point to each side of a given triangle. Here is an example of a pedal triangle.

Part 1.

• Construct a pedal triangle for an acute triangle.

• Point P is the intersection of the perpendiculars. Is point P always on the inside of the pedal triangle? Is it always on the inside of the original triangle? Explain your mathematical reasoning.

Part 2.

• Construct a pedal triangle for a right triangle and an obtuse triangle. Investigate point P for each triangle.

• Does your conclusions hold for a equilateral, isosceles, and scalene triangles?

7) Logo the Turtle

Part 1. Go onto the logo turtle applet, Math playground's logo applet, or Papert: logo. When making the different shapes, think about how logo is useful for teaching geometric concepts to students. Explain the any mathematical knowledge needed to construct these shapes in logo.

• Make a square, circle, octagon, and star.

• Now determine different ways to make the same objects.

• Make two different isosceles triangles.

• Make a triangle and its medial triangle.

• Make several regular polygons.

• Make the picture of the house shown below and provide the code.

• Make another figure with multiple shapes.

Part 2. Research the history of the logo turtle. Discuss your findings and any parts you found interesting (in general but also to mathematics).

8) Tangents part 2.

Make these constructions given two circles:

1) When the two circles are disjoint and not nested, construct lines that are tangent to both circles and explain how you did it.

or

2) When two circle are disjoint and nested, construct a circle that is tangent to both given circles and explain how you did it.

In the write-up reflect on your construction explaining the mathematics behind making the tangent lines and the tangent circle.