Handson Geometry Lessons with Right Triangles and Similarity

1
Hands-on Geometry Lessons with Right Triangles
and Similarity

• Jim Rahn
• LL Teach, Inc.
• www.llteach.com
• www.jamesrahn.com
• James.rahn_at_verizon.net

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Discovering the Pythagorean Theorem

• Use a pencil and ruler to draw the two diagonals
  in the two smallest squares on the top triangle
  on the Discovering the Pythagorean Theorem
  template.
• What type of figures are created by these
  diagonals?
• Place your Communicator on top of the template.
  Is it possible to fit these isosceles right
  triangles in the largest square? Show how this
  is possible on your Communicator.

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Discovering the Pythagorean Theorem

• Using the second triangle on the Discovering the
  Pythagorean Theorem template.
• Place your Communicator on top of the template.
• The one square has been divided into what shapes
  this time?
• Is it possible to fit these right triangles and
  the small square in the largest square? Show how
  this is possible on your Communicator. This is a
  little more challenging.

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Discovering the Pythagorean Theorem

• Using the top triangle If the area of each
  small square is 1, what is the area of the
  largest square?

2
1

• How long are the sides of the triangle?

1

• Using the bottom triangle If the area of each
  smallest square is 1 and the area of the largest
  square is 4, what is the area of the middle
  square?

4

• How long are the sides of the triangle?

1
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Discovering the Pythagorean Theorem

• If the length of the legs of a right triangle are
  a and b and the length of the hypotenuse is c.
  What is the area of each of the squares?

c2
c
a
a2
b
b2

• What do you know about the relationship of the
  areas?

c2
a

• Use the area of the squares to write a formula
  that describes the relationship that is true.

c
a2
b
b2
6

• Suppose a 2 and b 3, what do you know about
  the area of the triangles?
• What do you know about the length of c?

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• Suppose a 4 and c 10, what do you know about
  the area of the triangles?
• What do you know about the length of b?

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• Suppose b 3 and c 8, what do you know about
  the area of the triangles?
• What do you know about the length of a?

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Developing an Understanding for a Special Right
Triangle

• The large figure at the right is an equilateral
  triangle.
• What are the three properties of the line in the
  middle of the triangle?
• Label these properties on your Communicator.

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Developing an Understanding for a Special Right
Triangle

• Label the side of the equilateral triangle as 2.
• Find the length of all segments in the drawing
  and label them on your Communicator.
• Record the length of each leg and the hypotenuse
  on the chart.

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Developing an Understanding for a Special Right
Triangle

• Label the side of the equilateral triangle as 4.
• Find the length of all segments in the drawing
  and label them on your Communicator.
• Record the length of each leg and the hypotenuse
  on the chart.

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Developing an Understanding for a Special Right
Triangle

• Label the side of the equilateral triangle as 6.
• Find the length of all segments in the drawing
  and label them on your Communicator.
• Record the length of each leg and the hypotenuse
  on the chart.

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Developing an Understanding for a Special Right
Triangle

• What observations can you make from your chart?
  
• Where is the longest side of the triangle?
• Where is the shortest side of the triangle?
• What do you notice about the longer leg?

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Developing an Understanding for a Special Right
Triangle

• Find the length of the two legs in this right
  triangle at the right.

2a
a
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Developing an Understanding for a Special Right
Triangle

• Find the length of the two legs in this right
  triangle at the right.

16
8
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Developing an Understanding for a Special Right
Triangle

• Find the length of the two legs in this right
  triangle at the right.

10
5
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Developing an Understanding for a Special Right
Triangle

• Find the length of the two legs in this right
  triangle at the right.

14
7
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Developing an Understanding for a Special Right
Triangle

• Find the length of the two legs in this right
  triangle at the right.

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Developing an Understanding for Similar Polygons
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Working with Similar Polygons

• Place the Communicator on top of the Similar
  Trapezoids template.
• Trace the angles of the largest trapezoid.
• Slide the Communicator around so that you can
  compare these angles with the angles on each of
  the other trapezoids.
• What do you observe?

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Working with Similar Polygons

• Label the angles of the smallest trapezoid as 60
  and 120 degrees.
• Find the measure of each of the angles in the
  other three trapezoids.

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Working with Similar Polygons

• Clear the Communicator.
• Trace the second smallest trapezoid.
• Slide the Communicator on top of the smallest
  trapezoid and compare the corresponding sides.
• What do you observe?

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Working with Similar Polygons

• Compare each of the trapezoid to the smallest
  trapezoid.
• Describe the ratio of the corresponding sides for
  any two trapezoids.

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Working with Similar Polygons

• Label the lengths of the smallest trapezoid as
  1,1,1 and 2.
• Find the lengths of the all sides of all the
  other trapezoids.
• Find the perimeter of each trapezoid. What do
  you observe about their perimeters?

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Working with Similar Polygons

• Trace the next to the largest trapezoid on your
  Communicator.
• Slide your traced figure on top of the smallest
  trapezoid. How many smallest trapezoids fit in
  the traced trapezoid?

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Working with Similar Polygons

• Trace the smallest trapezoid on your
  Communicator. Into how many equilateral
  triangles can you subdivide it?
• Trace the next to the smallest trapezoid on your
  Communicator.
• Slide your traced figure on top of the smallest
  trapezoid. How many smallest trapezoids fit in
  the traced trapezoid? (You will have to subdivide
  one of the smallest trapezoids.)

What does this tell you about their area?
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Working with Similar Polygons

• Continue to fill each trapezoid with the smallest
  trapezoid to find out how their areas compare.
• Describe how the area of each trapezoid compares
  with the smallest trapezoid. How did their sides
  compare?
• Describe how the area any two trapezoids compare.
  How did their sides compare?

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Working with Similar Polygons

• Label the lengths of the smallest trapezoid as
  1,1,1 and 2.
• Find the lengths of the all sides of all the
  other trapezoids.
• Find the area of each trapezoid. Are your area
  ratios confirmed?

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Creating a Similar Polygon
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• Place the Four Quadrant Coordinate Labeled Graph
  (1/4 squares) Horizontal template in your
  Communicator.
• Graph the quadrilateral whose vertices are
  located at (4,2), (2, -2) (-2, -2) and (-4,2).
  What is the name of this quadrilateral?

• Find the length of each side, the perimeter and
  the area of the quadrilateral.

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• In your group create a second quadrilateral that
  is a dilation of the first quadrilateral whose
  vertices are (4a,2a), (2a, -2a) (-2a, -2a) and
  (-4a,2a) wherea1/2, 1 ½, 2 and 3.
• Compare your quadrilaterals to each other.

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• Find the length of each side, the perimeter and
  the area of the new quadrilateral.

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• Compare the sides of the four quadrilaterals.

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• Compare the perimeters of the four quadrilaterals.

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• Compare the areas of the four quadrilaterals.

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• What observations can you make?

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Hands-on Geometry Lessons with Right Triangles
and Similarity

• Jim Rahn
• LL Teach, Inc.
• www.llteach.com
• www.jamesrahn.com
• James.rahn_at_verizon.net