Download Similarity in Triangles

Math Investigations
Triangles
Name
Similarity
Two polygons are similar if one is the result of a dilation or contraction of the other. ratio of their
corresponding sides is a constant and their corresponding angles are congruent.
For triangles (and only for triangles), it is sufficient to know that all of their angles are congruent.
C
Z
h2
h1
X
Y
A
B
D
W
AA Similarity
If two angles of one triangle are congruent to two angles of another triangle, then they are similar
triangles.
Proof: Clearly if two pairs of angles are congruent, since the angles of a triangle add up to 180 degrees,
then the third pair of angles must be congruent.
SAS Similarity
If two sides of one triangle have the same ratio as the two sides of another triangle, and the angles
between theses sides are congruent, then they are similar triangles.
Proof:
XZ AC

and that X  A . Then
XY AB
h
h
sin( X )  1  2  sin( A)
XZ AC
Look at the figure above. Suppose that we know that
Also,
tan( X ) 
h1
h
 2  tan( A)
WX DA
By the segment addition postulate
h1 WX WY


h2 DA DB
So
tan(Y ) 
h1
h
 2  tan( B)
WY DB
Since both angles are between 0 and 180 degrees, B  Y which means sin( B ) 
h1
h
 2 , so
ZY BC
h1 YZ
, and the triangles are similar.

h2 BC
L Marizza A Bailey
Art of Problem Solving: Geometry
BASIS Scottsdale
Math Investigations
Triangles
Name
Exercises:
1. Let ABC be a triangle. Suppose M is on AB and N is on BC , so that MN
Show that ABC ~ MBN .
AC .
Step 1: Illustrate the triangle and the points.
Step 2: Can you tell if any angles are congruent? Explain your reasoning.
Step 3: Use one of the theorems or definitions to finish the proof.
2. Let ABC be a right triangle, with mB  90o. Suppose M is on AC so that mBMC  90o. ,
Show that ABC ~ BMC ~ AMB .
Proof:
Step 1: Illustrate the triangle and all the points.
Step 2: Draw the triangles separately so all the right angles are on the left base side.
Step 3: Show that all there are three pairs of congruent angles for each pair of triangles.
Step 4: Use the appropriate theorem.
L Marizza A Bailey
Art of Problem Solving: Geometry
BASIS Scottsdale
Math Investigations
Triangles
Name
3. Let line segment AB and CD intersect at the point M , and that AC BD
Show that AMC ~ BMD
Proof: ( I think you know the steps now).
Give each of these theorems a name. Share with the class and we'll choose the most popular name.
Problems:
1. Find RS, QR and TS
2. Find ON and MN
N
M
1.6
1.2
0
L Marizza A Bailey
Art of Problem Solving: Geometry
2
L
BASIS Scottsdale
Math Investigations
Triangles
Name
3. Find x in terms of y given the diagram below.
4. In triangle ABC, assume AC = 4, BC = 3, AB=5, and assume mACB  90o. The sequence of points,
Ci are generated by the rule that Ci 1 is the foot of the altitude Ci 1Ci . Calculate the sum

C C
i 0
L Marizza A Bailey
i
i 1
Art of Problem Solving: Geometry
BASIS Scottsdale
Math Investigations
Triangles
Name
5. In rectangle ABCD shown below, points F and G lie of the segment AB so that AF  FG  GB
and E is the midpoint of DC. Also AC intersects EF at H and EG at J . If the area of the rectangle
is 70, what is the area of the triangle AHF ?
E
D
J
H
A
C
F
B
G
6. Points P, Q, R, S, T, and U are on the sides of the triangle ABC, as shown. They are placed so that the
line segments UR, QT , and SP all pass through the point X, and are parallel to BC , CA, and AB
respectively. Show that
PQ RS TU


1
BC CA AB
A
S
T
U
B
L Marizza A Bailey
R
X
P
Q
Art of Problem Solving: Geometry
C
BASIS Scottsdale