Concept Byte: Exploring Proportions in Triangles
Use With Lesson 7-5
TECHNOLOGY
Activity 1
Use geometry software to draw
△
A
B
C
.
white up pointing triangle eh b c . Construct point D on
A
B
¯
.
eh b bar , . Next, construct a line through D parallel to
A
C
¯
.
eh c bar , . Then construct the intersection E of the parallel line with
B
C
¯
.
b c bar , .
- Measure
B
D
¯
,
D
A
¯
,
B
E
¯
,
and
E
C
¯
.
b d bar , comma , d eh bar , comma , b e bar , comma , and . e c bar , . Calculate the ratios
B
D
D
A
and
B
E
E
C
.
fraction b d , over d eh end fraction . and . fraction b e , over e c end fraction . .
- Manipulate
△
A
B
C
white up pointing triangle eh b c and observe
B
D
D
A
and
B
E
E
C
.
fraction b d , over d eh end fraction . and . fraction b e , over e c end fraction . . What do you notice?
- Make a conjecture about the four segments formed by a line parallel to one side of a triangle intersecting the other two sides.
Activity 2
Use geometry software to draw
△
A
B
C
.
white up pointing triangle eh b c . Construct the bisector of
∠
A
.
angle eh . Label the intersection of the bisector and
C
B
¯
c b bar with the letter D.
- Measure
A
C
¯
,
A
B
¯
,
C
D
¯
,
and
B
D
¯
.
eh c bar , comma , eh b bar , comma , c d bar , comma , and . b d bar , . Calculate the ratios
A
C
A
B
and
C
D
B
D
.
fraction eh c , over eh b end fraction . and . fraction c d , over b d end fraction . .
- Manipulate
△
A
B
C
white up pointing triangle eh b c and observe
A
C
A
B
and
C
D
B
D
.
fraction eh c , over eh b end fraction . and . fraction c d , over b d end fraction . . What do you notice?
- Make a conjecture about the two sides of a triangle and the two segments formed by the bisector of the opposite angle.
Exercises
-
Construct
A
B
↔
‖
C
D
↔
‖
E
F
↔
.
modified eh b with left right arrow above . double vertical bar , modified c d with left right arrow above , double vertical bar . modified e f with left right arrow above , . Then construct two transversals that intersect all three parallel lines. Measure
A
C
¯
,
C
E
¯
,
B
D
¯
,
and
D
F
¯
.
eh c bar , comma , c e bar , comma , b d bar , comma , and . d f bar , . Calculate the ratios
A
C
C
E
and
B
D
D
F
.
fraction eh c , over c e end fraction . and . fraction b d , over d f end fraction . . Manipulate the locations of A and B and observe
A
C
C
E
and
B
D
D
F
.
fraction eh c , over c e end fraction . and . fraction b d , over d f end fraction . . Make a conjecture about the segments of the transversals formed by the three parallel lines intersecting two transversals.
- Suppose four or more parallel lines intersect two transversals. Make a conjecture about the segments of the transversals.
