HSCE: G2.3.1 (need link here) Clarifying Examples and Activities

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HSCE: G2.3.1
Prove that triangles are congruent using the SSS, SAS, ASA,
and AAS criteria, and for right triangles, the hypotenuse
-
leg criterion.



Also elate to G 3.1.1 Isometries

(need link here)


Clarifying Examples and Activities
:


Students will learn
what the minimum conditions are to determine a triangle
uniquely. These conditions are found in the triangle congruence theorems,
which have been utilized from the time of ancient Greece.


Example 1:


Using fettucine and angle templates, students explore c
ongruence theorems
by creating counter
-
examples. Lesson is fully explained at
http://www.mste.uiuc.edu/courses/ci303fa01/students/kethoms1/
unitplan/day4.html


Extension:


To extend ideas in lesson, discuss the special case of a right triangle
(hypotenuse
-
leg postulate). Right triangle applications of congruence
theorems as well as the h
-
l postulate can be found at
http://library.thinkquest.org/20991/geo/crtri.html#HL


HSCE: G2.3.2
Use theorems about congruent triangles to prove additional
theorems and solve problems, with and without use of
coordinates.

Example 1:


Perplexing Parallelograms

A surprising result occurs when two line segments are drawn through a point
on the diagonal of a parallelogram and parallel to the sides. From this
construction, students are able to make various conjectures, and the basis of
this lesson is considering st
rategies for proving (or disproving) one of those
conjectures.

http://illuminations.nctm.org/LessonDetail.aspx?id=L709


HSCE: G2.3.3

Prove that triangles are similar by using SSS, SAS
, and AA
conditions for similarity.

Relate to G 3.1 (
need link here
)

Example 1:



This activity provides practice in proving similar triangles, finding unknown
lengths using similar triangles, and also demonstrates a practical application
for similar tria
ngles.


Materials:


Overhead projector

Ruler

Meter Stick

Butcher paper


Activity:


Lay butcher paper out on floor and position overhead projector so that a
meter stick placed on the edge of the paper will produce a shadow that can
be marked and measured
on paper. Place meter stick so that it is resting
against an object that will position it at a 90 degree angle from the floor.
Have students mark the end of the meter stick shadow. Replace meter stick
with a ruler and have students mark the end of the r
uler shadow. Measure
the length of each shadow. Have students sketch a representation and label
lengths and indicate congruent angles. Lead class discussion to determine if
the resulting triangles are similar. (Could conclude using AA or SAS
conditions
) Have students use their drawing and discussion to construct a
proof.


Extension:


Using knowledge gained from activity, have students work with a partner to
determine the height of the school, a post, etc. using shadows.



Example 2:




Given CB // ED, prove that ∆ABC and ∆ADE are similar triangles.


HSCE: G2.3.4

Use theorems about similar triangles to solve problems with
and without use of coordinates.


Example 1:


A brick building casts a shadow 120 feel long. At

the same time a 6ft man
casts a shadow of 15 ft. How tall is the building?


Example 2:


Explain why the mid
-
segment of a triangle (segment joining the midpoints of
2 sides
) is half the length of the third side.



Example 3:


Which of the following tria
ngles are always similar? Explain

a. Right triangles

b. Isosceles triangles

c. Equilateral triangles


Example 4:


Two ladders are leaned against a wall such that they make the same angle
with the ground. The 10' ladder reaches 8' up the wall. How much

further
up the wall does the 18' ladder reach?

1.

4.5'

2.

6.4'

3.

14.4'

4.

22.4'

http://www.regentsprep.org/Regents/math/similar/PracSim.htm


Enrichment:


The soccer stadium wall casts a shadow that extends 150 feet from

its base when the edge of the shadow forms
a 23
-
degree angle with the
ground. What is the height of the stadium wall to the nearest foot?


Web Resources:


Midpoints of successive sides of any quadrilateral form a Varignon
Parallelogram. Pierre Varignon’s (1654
-
1722) theorem was published
posthumou
sly in 1731. Have students discover and investigate why this
theorem works. Investigate with string or interactive geometry tool.

http://www.cut
-
the
-
knot.org/Curriculum/Geometry/Varignon.shtml


Discovering Geometry, Chapter 11, Similar Triangles
http://keymath.com/x19504.xml


Sierpinski Triangle, Fractals and Self
-
similarlty

Notice that if you magnify any of the small triangles, you get back to the
original big triangle! In other words, this object is self simi
lar. Self similarity
is an important property of fractal objects. Check this out by zooming into
the picture below.

http://math.rice.edu/~lanius/fractals/selfsim.htmlhttp


HSCE: G2.3.5
Know and apply the theorem stating that the effect of a scale
factor o
f k relating one two dimensional figure to another or one three
dimensional figure to another, on the length, area, and volume of the figures is to
multiply each by k, k
2
, and k
3
, respectively.

Example 1:


Growing Cubes Investigation: Students discover eff
ect of a scale factor k on
the perimeter, surface area, and volume of a cube.


Build or draw a set of cubes with the following edge lengths.

Be sure to identify the measure used.


Edge Length


Unit =

Perimeter of

One face

Unit =

Surface Area of

The cube

Unit =

Volume of Cube


Unit =

E

P

S

V

1




2




3




4




10




N





Describe the patterns in the tables, graphs, and equations that relate



(Edge length, perimeter)






(Edge length, surface area)




(Edge length, volume of cube)


When edge lengt
h E is multiplied by k, explain how



the perimeter of one face changes?



the surface area of the cube changes?



the volume of the cube changes?


Example 2:


There are 250 paperclips in a rectangular box. If doubling all of the
dimensions of the original

box makes a jumbo box, how many paperclips will
the jumbo box hold?


Answer: The dimensions of the box are multiplied by a factor of k, thus the
volume is multiplied by
. The jumbo box will then hold
papercli
ps.


Web Resources:



This two
-
part example illustrates how students can learn about the length,
perimeter, area, and volume of similar objects using dynamic figures. In this
part, Side Length and Area of Similar Figures, the user can manipulate the
side

lengths of one of two similar rectangles and the scale factor to learn
about how the side lengths, perimeters, and areas of the two rectangles are
related.

http://standards.
nctm.org/document/eexamples/chap6/6.3/index.html




Students will use an applet to explain the relationship between the volume of
two similar rectangular prisms and the scale factor

http://illuminations.nctm.org/LessonDetail.aspx?ID=L261