What can you say about the ratios of consecutive sides looking at the table you made in Part B?
A
a
See solution.
B
b
See solution.
C
c
The proportion of the segments created by the angle bisector of a triangle is equal to the proportion of their respective consecutive sides.
Practice makes perfect
a
Let's start with drawing three triangles, one acute, one right, and one obtuse. We will label them ABC, MNP and WXY respectively.
The next step will be to draw angle bisector BD. To do this we will start with putting a compass at point B and drawing an arc that intersects both BA and BC.
Let's label the points of intersection Q and R.
With the compass at point Q, draw an arc in the interior of the angle.
Keeping the same compass setting, place the compass at point R and draw an arc that intersects the arc drawn in the previous step.
Next, we will connect the point of intersection of arcs with the point B. Finally, we will expand this segment to intersect AC. Label this point D.
Now, we will find angle bisectors NQ and XZ in the same way.
b
In this part we are asked to copy and complete the given table with the appropriate values. To do this, we will measure appropriate segments using a ruler. Let's start with the first triangle.
Now we will measure the rest of segments.
Finally, we will complete the given table.
Triangle
Length
Ratio
ABC
AD
1.3
AD/CD
1.3/1.2=1.08
CD
1.2
AB
2.9
AB/CB
2.9/2.69=1.08
CB
2.69
MNP
MQ
1.39
MQ/PQ
1.39/2.61=0.53
PQ
2.61
MN
2.5
MN/PN
2.5/4.72=0.53
PN
4.72
WXY
WZ
2.94
WZ/YZ
2.94/2.56=1.15
YZ
2.56
WX
3.35
WX/YX
3.35/2.92=1.15
YX
2.92
c
Looking at the table we made in Part B, we can see that the ratios are equal for each triangle. Therefore, we can assume that the proportion of the segments created by the angle bisector of a triangle is equal to the proportion of their respective consecutive sides.
