**Objectives:** To make conjectures about the sides,
perimeters, and areas of one triangle

and the triangle that is constructed from the midpoints of the sides of the
original triangle.

**Materials:** Geometer’s Sketchpad, overhead projector for computer (this
could be easily

adapted for the TI-92), class set of computers with Geometer’s Sketchpad on
them.

**Role of Technology: **We will use the Geometer’s Sketchpad to construct
triangles and

their midpoints. Sketchpad will also calculate the length of the sides of the
triangles, the

perimeter and the area for us. Constructing various triangles on Geometer’s
Sketchpad

will allow us to make conjectures that we can apply to all triangles.

**Previous knowledge base:** Students have prior knowledge of triangles,
length,

midpoint, perimeter, and area. Students also have some experience with the
Geometer’s

Sketchpad.

**Lesson:**

• Set Establishment (5 - 7 minutes)

Tell the student that we are going to work with Geometer’s Sketchpad today to
make

important discoveries regarding triangles. Ask them if they recall how to find
the

midpoint of a line. The midpoint is the actual “middle point” of a line such
that the

length from the midpoint to each endpoint is the same. The formula to find the
midpoint

of the line with endpoints P_{1}(a,b) and P_{2}(c,d) is
. Next ask the

students how to find the perimeter and the area of a triangle. To find the
perimeter, add

the length of the three sides. Area of a triangle is
(base)(height). We will not be

computing these formulas by hand today, we will let Geometer’s Sketchpad compute

them for us.

• Creating the First Triangle (3 minutes)

Open up Geometer’s Sketchpad. While performing the following procedures, tell

your students what you are doing explicitly so that they can follow along.
Choose the

point (it looks like a dot on the left side of the screen). Make three points on
the

screen. Now choose the arrow and select all three points by holding down SHIFT
and

clicking on each point. Now choose SEGMENT from the CONSTRUCT menu. We

have just constructed a triangle. Ask if there are any questions.

• Constructing a Second Triangle from the Midpoints of the First (7 minutes)

Now, using the arrow again, select the three sides of the triangle we have
constructed.

From the CONSTRUCT menu again, choose POINT AT MIDPOINT. A point

appears at the midpoint of each segment. Now with the SHIFT key held down,
select

our three midpoints. Then, choose SEGMENT from the CONSTRUCT menu. We

have just constructed a smaller triangle inside our original triangle using the

midpoints of the first triangle as the vertices of the second. Ask if there are
any

questions and quickly walk around the room to make sure everyone has the right

picture.

• Analyzing the Measures of the Angles and Sides (15 minutes)

Using the “hand” on the left toolbar, label your points.
This will make it easier to

analyze our measurements. Now use the arrow again and select the sides of the
first

triangle. Then, choose LENGTH from the MEASURE menu. This calculates the

length of the sides of the triangle. Repeat this for the second triangle. Ask
the

students if they notice any relationship between the length of the sides of the
larger

triangle and the length of the sides of the smaller triangle. If they cannot
come up

with any conjectures, tell them to specifically look for a relationship between
the

length of the side of the larger triangle and the length of the side of the
smaller

triangle that is parallel to it. For example, in the picture below the students
should

look for a relationship between side AC and side EF. The students should notice
that

the length of the each side of the larger triangle is
twice the length of the side of the

smaller triangle that is opposite it (and also parallel to it). So in this
example, AC is

twice as long as EF, AB is twice DF, and CB is twice DE. Now let’s look at the

measures of the angles in each triangle. To measure the angle, select the three
points

that define the angle in order. For example, if I wanted to measure angle A in
the

picture above, I would select point C, then A, then B (all while holding the
SHIFT

button as before). Then choose ANGLE from the MEASURE menu. Another way to

get this same angle would be to choose points D, A, and E since ∠ CAB is equal
to

∠ DAE. Repeat this procedure for the angles at the vertices of each triangle.
There

should be a total of 6 angle measures. For review, ask the students what the sum
of

the three angles in each triangle should be (180°). Once the students have
calculated

the measures of all the angles, ask if they notice anything of interest about
the angles.

The students should notice that the opposite angles are equal. In our example
above,

∠ A = ∠ F, ∠ B = ∠ D, and ∠ C = ∠ E.

• Analyzing the Perimeters and Areas of the Triangles (15 minutes)

Now we will look at the perimeter and area of each of our triangles. Select the
three

vertices of the larger triangle. Then choose POLYGON INTERIOR from the

CONSTRUCT menu. This should color the interior of the triangle. Now choose

PERIMETER and AREA (one at a time) from the MEASURE menu. Repeat for the

smaller triangle. We will now compare the areas and perimeters of the two
triangles.

It may be difficult to understand what is happening here since the students will
only

have two measurements to compare for the perimeter and two to compare for the

area. Make a table on the board and ask students to give you their information
to fill

the chart. With a lot of data from the class, the students should be able to see
the

relationship between the triangles. A sample table is below (you may need more
than

4 entries).

Perimeter (Big Triangle) |
Perimeter (Small Triangle) |
Area (Big Triangle) |
Area (Small Triangle) |

From the table, the students should conclude that the
perimeter of the larger triangle

is twice the perimeter of the smaller triangle and that the area of the big
triangle is 4

times the area of the small triangle. It would be good for the students to drag
the

triangles around a bit to see that this really works.

• Assignment (5 minutes)

Hand out the following worksheet for homework and ask the class if there are any

further questions.

In the triangle below, each interior triangle is
constructed from the midpoints of the

triangle it is inside. You are given the following measurements.

Calculate the following lengths.

1. AD

2. LJ

3. QP

4. HK

Calculate the following angles.

5. ∠ CAB

6. ∠ QPR

7. ∠ NOM

8. ∠ KLJ

9. ∠ IHG

Calculate the following perimeters.

10. ABC

11. GHI

12. QPR

Calculate the following areas.

13. ABC

14. QPR

15. JKL