2.5 Relationships Within Triangles
Midsegments of a Triangle
Don’t Forget the^12 - In this section there are two types of relationships that the students need to keep in mind when
writing equations with variable expressions. The first involves the midpoint. When the expressions represent the two
parts of a segment separated by the midpoint they just have to set the expressions equal to each other. The second
is when comparing the length of a side of the triangle with the midsegment parallel to it. In this case they need to
multiply the expression representing the side of the triangle by^12 , and then set it equal to the expression representing
the midsegment. They may forget the^12 or forget to use parenthesis and distribute. Remind them that they need to
multiply the entire expression by^12 , not just the first term.
Additional Exercises:
- The proof given in the text of the Midsegment Theorem is a paragraph proof. Write the second part of the proof
as a two-column proof.
Answer: Refer to the triangle used in the text proof on the top of page 267.
TABLE2.8:
Statement Reason
AB,CB, andACare midsegments of 4 XY Z Given
ABis parallel toXY Midsegment Theorem (1)(^6) BAC∼= (^6) XCA Alternate Interior Angle Theorem
CBis parallel toX Z Midsegment Theorem (1)
(^6) X AC∼= (^6) BCA Alternate Interior Angle Theorem
AC∼=AC Reflexive Property of Congruence
4 AXC∼= 4 CBA ASA Postulate
AB∼=XC Definition of Congruent Triangles
Cis the midpoint ofXY Definition of Midsegment
XC=CY Definition of Midpoint
XY=XC+CY Segment Addition Postulate
XY=XC+XC Substitution Property of Equality
AB=XC Definition of Congruent Segments
XY=AB+AB= 2 AB Substitution Property of Equality
AB=^12 XY Division Property of Equality
Perpendicular Bisectors in Triangles
Construction Frustrations –Using a compass and straightedge to make clean, accurate constructions takes a bit of
practice. Some students will pick up the skill quickly and others will struggle. What is nice about doing construction
in the classroom is that it is often the students that typically struggle with mathematics, the more artistically minded
students that excel and learn from constructing figures.
Chapter 2. Geometry TE - Common Errors