# Point at Midpoint

Electronics - Devices

Oct 10, 2013 (4 years and 9 months ago)

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Exploring Similar Triangles Part 1 Using Geometer’s Sketchpad

1. Open a new sketch window in Sketchpad. Plot three points anywhere within the
sketch window. Hold down the shift key and select all three points. Wi
th all three
points selected, construct segments connecting these points. You should now have
a triangle with vertices labeled A, B, and C. If not, select the vertex labels and
rename them. See Figure 1.

2. Using the pointer select sides AB and AC. Constr
uct the midpoint of each
segment using the
Point at Midpoint

command. With both of these points selected,
construct the segment connecting them. Drag any one of the five labeled points.
What do you notice about the shape of triangle ADE and triangle ABC? W
hat two
things do you notice about segment DE in relation to segment BC? How can you
confirm these observations? Write two possible theorems that results from these
observations. See Figure 1.

3. Select points A and D. Measure the
distance

between these t
wo points. Repeat
this for points A and E, points D and E, points A and B, points A and C, and points
B and C. What is the ratio of the length of each side of triangle ADE to the
corresponding side of triangle ABC? For example what is (length AD)/(length
A
B)? Use the calculate command to record these three ratios. What happens to the
distances between points as you drag the triangle? What happens to the value of
the ratios as you drag the triangle? See Figure 2.

4. Select points A, D, and E and
construct p
olygon interior
. With the interior of
triangle ADE selected, measure both the
perimeter

and
area

Repeat this procedure for triangle ABC. (You may want to change the color of one
triangle interior.) See Figure 2.

5. Calculate the ratio of
the perimeter of triangle ADE to the perimeter of triangle
ABC. Does this ratio change as you drag the triangle? Compare this ratio to the
ratio of the corresponding side lengths. What do you notice? Write a possible
theorem that explains the ratio of the
two perimeters with respect to this
construction.

6. Calculate the ratio of the area of triangle ADE to the area of triangle ABC. How
does this ratio compare to the ratio of the perimeters and the ratio of the
corresponding sides? Write a possible theorem

that explains the ratio of the two
areas with respect to this construction.

7. Based on the theorems you conjectured in parts 5 and 6, if points F and G were
constructed to be the midpoints of segments AD and AE respectively, what would
the ratio of the
corresponding sides of triangles AFG and triangle ABC be? What
would the ratio of their perimeters be? What would the ratio of the areas of these
two triangles be? Test your conjectures by constructing points F and G, the interior
of triangle AFG and calcu
lating the desired ratios.

8. Were your conjectures shown to be correct? If not, how should they be
modified?

9. Assume points H and I were positioned one
-
third of the way from point A to
points B and C respectively. What should the values of the followi
ng ratios be:

AH / AC, (Perimeter AHI )/ (Perimeter ABC), (Area AHI) / (Area ABC)

10. Write a theorem for the general case of any two similar triangles relating the
ratios of the corresponding sides to the ratio of the perimeters and the ratio of the
area
s.

Figure 1

Figure 2