Parallel Lines and Proportional Parts

• Write the three ratios of the sides given the two similar triangles.

4

105

2

C

A E

B D

• It is also true that the two parts of each side are proportional.

• Triangle Proportionality: If a line is parallel to one side of a triangle and intersects the other two sides, then it separates these sides into segments that are proportional.

• This means, we could write another set of equal ratios…

4

105

2

C

A E

B D

• Solve for x and y.

y

6x

2.5

8 3B

E

D

A C

• Given the picture, if M and N are midpoints, find the value of x.

x

3

5

NM

B

A

C

• This shows us the next property about similar triangles.

• Thm 7.6: If you connect the midpoints of two sides of a triangle, then the length of that segment is ½ the length of the third side of the triangle.

• Ex. In ∆ABC, M is the midpoint of AB, N is the midpoint of BC and P is the midpoint of AC. Find the perimeter of ∆MNP if AB = 16, BC = 18 and AC = 22.

• Corollary: If three or more parallel lines are cut by two transversals, then they are cut proportionally.

E

F

G

D

C

B

A

AB

BC=?

GF

BC

GF=CD

?

?

AD=AG

AE

AF

?=AC

CD