USING SIMILARITY THEOREMS

THEOREM S

THEOREM 8.2 Side-Side-Side (SSS) Similarity Theorem

If the corresponding sides of two triangles are proportional, then the triangles are similar.

If = =A BPQ

BCQR

CARP

then ABC ~ PQR.

A

B C

P

Q R

USING SIMILARITY THEOREMS

THEOREM S

THEOREM 8.3 Side-Angle-Side (SAS) Similarity Theorem

If an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar.

then XYZ ~ MNP.

ZXPM

XYMN

If X M and =

X

Z Y

M

P N

Proof of Theorem 8.2

GIVEN

PROVE

= = STMN

RSLM

TRNL

RST ~ LMN

SOLUTION

Paragraph Proof

M

NL

R T

S

P Q

Locate P on RS so that PS = LM.

Draw PQ so that PQ RT.

Then RST ~ PSQ, by the AA Similarity Postulate, and .= = ST SQ

RS PS

TR QP

Use the definition of congruent triangles and the AA Similarity Postulate to conclude that RST ~ LMN.

Because PS = LM, you can substitute in the given proportion and find that SQ = MN and QP = NL. By the SSS Congruence Theorem,

it follows that PSQ LMN.

E

F D8

6 4A C

B

12

6 9

G J

H

14

6 10

Using the SSS Similarity Theorem

Which of the following three triangles are similar?

SOLUTION To decide which of the triangles are similar, consider the

ratios of the lengths of corresponding sides.

Ratios of Side Lengths of ABC and DEF

= = , 6 4

AB DE

3 2

Shortest sides

= = , 12 8

CA FD

3 2

Longest sides

= = 9 6

BC EF

3 2

Remaining sides

Because all of the ratios are equal, ABC ~ DEF

= = ,1214

CA JG

67

Longest sides

E

F D8

6 4

Using the SSS Similarity Theorem

A C

B

G J

H

Which of the following three triangles are similar?

SOLUTION To decide which of the triangles are similar, consider the

ratios of the lengths of corresponding sides.

Ratios of Side Lengths of ABC and GHJ

12 14

6 6 109

= = 1, 6 6

AB GH

Shortest sides

= 910

BC HJ

Remaining sides

Because all of the ratios are not equal, ABC and DEF are not similar.

E

F D8

6 4A C

B

12

6 9

G J

H

14

6 10

Since ABC is similar to DEF and ABC is not similar to GHJ, DEF is not similar to GHJ.

Using the SAS Similarity Theorem

Use the given lengths to prove that RST ~ PSQ.

SOLUTION PROVE RST ~ PSQ

GIVEN SP = 4, PR = 12, SQ = 5, QT = 15

Paragraph Proof Use the SAS Similarity Theorem. Find the ratios of the lengths of the corresponding sides.

= = = = 4 SR SP

16 4

SP + PR SP

4 + 12 4

= = = = 4 ST SQ

20 5

SQ + QT SQ

5 + 15 5

Because S is the included angle in both triangles, use the SAS Similarity Theorem to conclude that RST ~ PSQ.

The side lengths SR and ST are proportional to the corresponding side lengths of PSQ.

12

4 5

15

P Q

S

R T

USING SIMILAR TRIANGLES IN REAL LIFE

SCALE DRAWING As you move the tracing pin of a pantograph along a figure, the pencil attached to the far end draws an enlargement.

Using a Pantograph

P

R

T

S

Q

USING SIMILAR TRIANGLES IN REAL LIFE

Using a Pantograph

As the pantograph expands and contracts, the three brads and the tracing pin always form the vertices of a parallelogram.

P

R

T

S

Q

USING SIMILAR TRIANGLES IN REAL LIFE

Using a Pantograph

The ratio of PR to PT is always equal to the ratio of PQ to PS. Also, the suction cup, the tracing pin, and the pencil remain collinear.

P

R

T

S

Q

You know that . Because P P, you can apply the

SAS Similarity Theorem to conclude that PRQ ~ PTS.

= PQPS

PRPT

Using a Pantograph

S

R

Q T

P

How can you show that PRQ ~ PTS?

SOLUTION

SOLUTION Because the triangles are similar, you can set up a proportion to find the length of the cat in the enlarged drawing.

Using a Pantograph

10"

2.4"10"

S

R

Q T

P

= RQTS

PRPT

Write proportion.

=1020

2.4TS

=TS 4.8

Substitute.

Solve for TS.

So, the length of the cat in the enlarged drawing is 4.8 inches.

In the diagram, PR is 10 inches and RT is 10 inches. The length of the cat, RQ, in the original print is 2.4 inches. Find the length TS in the enlargement.

Finding Distance Indirectly

Similar triangles can be used to find distances that are difficult to measure directly.

ROCK CLIMBING You are at an indoor climbing wall. To estimate the height of the wall, you place a mirror on the floor 85 feet from the base of the wall. Then you walk backward until you can see the top of the wall centered in the mirror. You are 6.5 feet from the mirror and your eyes are 5 feet above the ground.

85 ft6.5 ft

5 ft

A

B

C E

DUse similar triangles to estimate the height of the wall.

Not drawn to scale

Finding Distance Indirectly

85 ft6.5 ft

5 ft

A

B

C E

D

Use similar triangles to estimate the height of the wall.

SOLUTION

Using the fact that ABC and EDC are right triangles, you can apply the AA Similarity Postulate to conclude that these two triangles are similar.

Due to the reflective property of mirrors, you can reason that ACB ECD.

85 ft6.5 ft

5 ft

A

B

C E

D

DE65.38

Finding Distance Indirectly

Use similar triangles to estimate the height of the wall.

SOLUTION

= ECAC

DEBA

Ratios of lengths of corresponding sides are equal.

Substitute.

Multiply each side by 5 and simplify.

DE5

= 856.5

So, the height of the wall is about 65 feet.