Recommend that students invest in a decent compass. They can get a quality, metal compass with some weight
behind it for less that $20. The $2 variety often does not hold the pencil steady.They slip, and are extremely
frustrating.
Here are some other tips for good construction: (1) Hold the compass at an angle. (2) Try rotating the paper while
holding the compass steady. (3) Work on a stack of a few papers so that the needle of the compass can really dig
into the paper and will not slip.
Perpendicular Bisector Quirks –There are two key ways in which the perpendicular bisector of a triangle is
different form the other segments in the triangle that the students will learn about in subsequent sections. Since
they are learning about the perpendicular bisector first these differences do not become apparent until the end of the
chapter.
The perpendicular bisector of the side of a triangle does not have to pass through a vertex. Have the students explore
in what situations the perpendicular bisector does pass through the vertex. They should discover that this is true for
equilateral triangles and for the vertex angle of isosceles triangles.
The point of concurrency of the three perpendiculars of a triangle, the circumcenter, can be located outside the
triangle. This is true for obtuse triangles. The circumcenter will be on the hypotenuse of a right triangle. This is also
true for the orthocenter, the point of concurrency of the altitudes.
Same Construction for Midpoint and Perpendicular Bisector –The Perpendicular Bisector Theorem is used to
construct the perpendicular bisector of a segment and to find the midpoint of a segment. When finding the midpoint,
the students should make the arcs, one from each endpoint with the same compass setting, to find two equidistant
points, but instead of drawing in the perpendicular bisector, they can just line up their ruler and mark the midpoint.
This will keep the drawing from getting overcrowded and confusing.
Angle Bisectors in Triangles
Check with a Third –When constructing the point of concurrency of the perpendicular bisectors or angle bisectors
of a triangle, it is strictly necessary to construct only two of the three segments. The theorems proved in the texts
ensure that all three segments meet in one point. It is advisable to construct the third segment as a check of accuracy.
Sometimes the compass will slip a bit while the student is doing the construction. If the three segments form a little
triangle, instead of meeting at a single point, the student will know that their drawing is not accurate and can go back
and check their marks.
Inscribed Circles –For a circle to be inscribed in a triangle, all three sides of the triangle must be tangent to the
circle. A tangent to a circle intersects the circle in exactly one point. After accurately finding the incenter, students
may have a difficult time finding the correct compass setting that will construct the inscribed circle. The best method
is to place the center of the compass at the incenter, choose one side, and adjust the compass setting until the compass
brushes by that side of the triangle, without passing through it. The word tangent does not have to be introduced
at this point if the students already have enough vocabulary to learn. When the incenter is correctly placed, the
compass should also hit the other two sides of the triangle once, creating the inscribed circle.
Additional Exercises:
- Construct an equilateral triangle. Now construct the perpendicular bisector of one of the sides. Construct the
angle bisector from the angle opposite of the side with the perpendicular bisector. What do you notice about these
two segments? Will this be true of a scalene triangle?
Answer: The segments should coincide on the equilateral triangle, but not on the scalene triangle.
- Construct an equilateral triangle. Now construct one of the angle bisectors. This will create two right triangles.
Label the measures of the angles of the right triangles. With your compass compare the lengths of the shorter leg to
the hypotenuse of either right triangle. What do you notice?
2.5. Relationships Within Triangles