Geometry in the Trees
Notes to the Teacher
Side

Splitter, Triangular Proportionality,
Medians, and Triangle Angle Bisector Theorems
Some Theorems you may wish to review and/or introduce prior to doing this exercise are:
Theorem 17 Isosceles Triangle The
orem
If two sides of a triangle are congruent, then the angles opposite those sides are congruent.
Theorem 18 Converse of the Isosceles Triangle Theorem
If two angles of a triangle are congruent, then the sides opposite those angles are congruent.
Theore
m 19
The bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base.
Theorem 21 Triangle Midsegment Theorem
If a segment joins the midpoints of two sides of a triangle, then the segment is parallel to the third
side an
d is half its length.
Theorem 22 Perpendicular Bisector Theorem
If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of
the segment
Theorem 23 Converse of the Perpendicular Bisector Theorem
If a point is equ
idistant from the endpoints of a segment, then it is on the perpendicular bisector of
the segment.
Theorem 24 Angle Bisector Theorem
If a point lies on the bisector of an angle, then the point is equidistant from the sides of the angle.
Theorem 25 Conve
rse of the Angle Bisector Theorem
If a point in the interior of an angle is equidistant from the sides of the angle, then the point is on
the angle bisector.
Theorem 26
The perpendicular bisectors of the sides of a triangle are concurrent at a point equid
istant from
the vertices.
Theorem 27
The bisectors of the angles of a triangle are concurrent at a point equidistant from the sides.
Theorem 28
The medians of a triangle are concurrent at a point that is 2/3 the distance from each vertex to the
midpoint
of the opposite side.
Theorem 44 Side

Splitter Theorem
If a line is parallel to one side of a triangle and intersects the other two sides, then it divides those
sides proportionally.
Theorem 45
If three parallel lines intersect two transversals, then th
e segments intersected on the transversal
are proportional.
Theorem 46
If a line divides two sides of a triangle proportionally, then it is parallel to the third side.
Theorem 47 Triangle angle Bisector Theorem
If a ray bisects an angle of a triangle,
then it divides the opposite side into two segments that are
proportional to the other two sides of the triangle.
Theorem 48 Perimeter and Areas of Similar Triangles
If the similarity ratio of two similar triangles is a/b, then:
a. the ratio of their pe
rimeters is a/b, and
b. the ratio of their areas is a squared to b squared.
Geometry in the Trees
Side

Splitter, Triangular Proportionality,
Medians, and Triangle Angle Bisector Theorems
1. Use the Side

Spl
itter Theorem to find x, given that
PQ
<
BC
.
2. Given:
PQ
<
BC
. Find the length of
AQ
.
3. Given
AE
<
BD
, solve for x.
4. Given:
OP
<
NQ
. Find the value of x and y.
Solve for x.
8.
LO
bisects
S
NLM, LM = 18, NO = 4 and LN = 10. Find OM
9
. Find x to the nearest tenth.
10. An angle bisector of a triangle divides the opposite side of the triangle into segments 8cm
and 4 cm long.
A
second side of the triangle is 4.4cm long. find all possible lengths of the third
side of the triangle.
Round answers to the nearest tenth of a centimeter.
11.
T
ABC ~
T
DEF. AB = 8 and DE = 14. Give the ratio of the perimeters and the ratio of the
areas.
12.
T
PIT ~
T
WVU. PI = 40 ft and WV = 35 ft. The area of
T
PIT is 1589 square feet.
W
hat is the
area of
T
WVU?
13. In
T
ACE, B is the midpoint of
AC
, D is the midpoint of
CE
and F is the midpoint of
AE
.
Given: AB = 12, BF = 13, and AF = 14.
S
ketch a diagram and
find each of the following.
a.
BC
b. AC
c. CD
d. DE
e. CE
f. FE
g. AE
h. BD
i. FD
j. Perimeter of
T
ACE
k. Perimeter of
T
BFD
l. Perimeter of
T
ABF
m. Perimeter of
T
BCD
n. Perimeter of
T
FDE
1
4
. Four
scouts
are trying to find t
he distance across the lake. They position themselves as
shown
. Al
ice
(A)
uses her compass to instruct Ch
ris (C)
and D
avid
(D)
to move along the line
they form with B
renda
(B)
until she sees
,
from her perspective
,
the angle between B
renda
and
Ch
ris
is eq
ual to the angle formed between Ch
ris
and D
avid
. They measure the distance
between Ch
ris
and B
renda
to be 35m, between Ch
ris
and D
avid
to be 46m, and between A
lice
and D
avid
to be 100m. How long is the lake from east to west? (Round answer to the neares
t
tenth of a meter)
Geometry in the Trees
Side

Splitter, Triangular Proportionality,
Medians, and Triangle Angle Bisector Theorems
Answers
1. 12
2. 9
3. 7
6
7
4. x = 10, y = 25.5
5. 5
6. 8
9. 14.4
10. 8.8
cm, 2.2cm
11. 4:7, 16:49
12. 1390.375
13.
a. 12
b. 24
c. 13
d. 13
e. 26
f. 14
g. 28
h. 14
i. 12
j. 78
k. 39
l. 39
m. 39
n. 39
14. 76.1m
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