Proportions and Similar Triangles

GeometryUnit 11, Day 8

Ms. Reed

Proportions and Similar Triangles

We will be investigating ways proportional relationships in triangles

You will need: Paper Ruler Protractor Calculator

On your paper:

1. Construct a triangle, label it ABC2. Create a line parallel to AC. Call the

intersection point on AB, D and the point on BC, E.

3. Measure DB, DA, BE, and EC4. Compare the ratios of BD/DA and BE/EC

5. WHAT DO YOU NOTICE?

Conclusion If a line is parallel to one side of

the triangle and intersects the other two sides, then it divides those sides proportionally.

This is called the Side-Splitter Theorem

Example 1 Set up the proportion

x =8

x

10

5

16

Example 2 Solve for x

x = 1.5

3

x

5

2.5

On your paper:

1. Create 3 Parallel Line2. Draw 2 transversals through the

lines so it looks like this:

3. Label as shown

a

b

c

d

What do you notice? Measure a, b, c and d. Compare the relationship between

a/b and c/d. WHAT DO YOU NOTICE?

What we discovered! If 3 parallel lines intersect two

transversals, then the segments intercepted on the transversals are proportional.

Example 3

16.5y

15 25

x

30 x =18 y = 27.5

Sail Making! When making a boat sail, all

of the seams are parallel. Find the missing variables

x = 2 ft, y=2.25 ft

2ft

2ft

3ft1.5ft

1.5ft1.5ft

x y

On your paper Create a new triangle and label it ABC Measure A Bisect A by drawing an angle with

half its measure. Label the intersection point with the

CB and the bisecting line point D Compare the ratios of CD/DB and CA/BA

WHAT DID YOU NOTICE?

Conclusion 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.

This is called the Triangle-Angle-Bisector Theorem.

Example 4 Set up the proportion

x=9.6

8

5x

6S

P

R

Q