Answer: The hypotenuse should be twice the length of the shorter leg.
Medians in Triangles
Vocabulary Overload –So far this chapter has introduced to a large number of vocabulary words, and there will
be more to come. This is a good time to stop and review the new words before the students become overwhelmed.
Have them make flashcards, or play a vocabulary game in class.
Label the Picture –When using the Concurrency of Medians Theorem to find the measure of segments, it is helpful
for the students to copy the figure onto their paper and write the given measures by the appropriate segments. When
they see the number in place, it allows them to concentrate on the relationships between the lengths since they no
longer have to work on remembering the specific numbers.
Median or Perpendicular Bisector –Students sometimes confuse the median and the perpendicular bisector since
they both involve the midpoint of a side of the triangle. The difference is that the perpendicular bisector must be
perpendicular to the side of the triangle, and the median must end at the opposite vertex.
Key Exercises:
- In what type(s) of triangles are the medians also perpendicular bisectors? Is this true of all three medians?
Answer: This is true of all three medians of an equilateral triangle, and the median that intersects and vertex angle
of an isosceles triangle.
Applications –Students are much more willing to spend time and effort learning about topics when they know of
their applications. Questions like the ones below improve student motivation.
Key Exercises:
In the following situations would it be best to find the circumcenter, incenter, or centroid?
- The drama club is building a triangular stage. They have supports on all three corners and want to put one in the
middle of the triangle.
Answer: Centroid, because it is the center of mass or the balancing point of the triangle
- A designer wants to fit the largest circular sink possible into a triangular countertop.
Answer: Incenter, because it is equidistant from the sides of the triangle.
Altitudes in Triangles
Extending the Side –Many students have trouble knowing when and how to extend the sides of a triangle when
drawing in an altitude. First, this only needs to be done with obtuse triangles when drawing the altitude that intersects
the vertex of one of the acute angles. It is the sides of the triangle that form the obtuse angle that need to be extended.
The students should rotate their paper so that the vertex of the acute angle they want to start an altitude from is above
the other two, and the segment opposite of this vertex is horizontal. Now they just need to extend the horizontal side
until it passes underneath the raised vertex.
The Altitude and Distance –The distance between a point and a line is defined to be the shortest segment with
one endpoint on the point and the other on the line. It has been shown that the shortest segment is the one that is
perpendicular to the line. So, the altitude is the segment along which the distance between a vertex and the opposite
side is measured. Seeing this connection will help students remember and understand why the length of the altitude
is the height of a triangle when calculating the triangle’s area using the formulaA=^12 bh.
Chapter 2. Geometry TE - Common Errors