Name ________________________
Worksheet 5.3
Ratios
Let’s consider the following scenario regarding triangle ACE: a) If B is the midpoint of CA and D is the midpoint of CE , show that

CBD ~

CAE . This is known as the Midsegment Theorem .
C
D
B b) If BD has a length of 7, what is the length of AE ?
A c) Can you establish that BD AE ? Justfiy. d) If we consider drawing in a parallel (i.e. BD ) to a third side of a triangle (i.e. AE ) – not necessarily through the midpoints of two sides – we will still have similar triangles. Explain. e) Give a clear statement of the Midsegment Theorem, in your own words.
E

This leads us to an interesting theorem…
Theorem: If there exists a series of parallel lines and they cut several transversals into parts, then the ratios of these parts will always be the same. a c
_____________________________ b d
Conclusion:
Another theorem involving proportions…
Theorem: An angle bisector will split the opposite side of a triangle into parts with the same ratio as the ratio between the two sides. b a c d
Conclusion:
____________________________
Definitions:
The ratio of a to b is the number a b
.
Ratios are sometimes written in the form a : b . This is not the same as “odds,” which are sometimes used when finding the specific probability of an even, as in gambling.
A proportion is an equation stating that two ratios are equal: a b
Useful fact: All of these are equivalent: a b
 c d b a
 d c a c
 b d
 c d d b
.
 c a ad
 bc .

The last one is the rule of cross-multiplication , a good rule for solving proportions, or determining if proportions are equivalent. It is important to understand that if the cross-products are equal , then the proportion is true (ratios are equal) .
A surprising fact: If a b
Prove this fact:
 c d
then a
 b b
 c
 d d
(we should be able to prove this!)
Exercises:
1. Which of these statements are true? Justify. a.
3
5

9
25
b. x y

6 x
6 y c. x

5 y

5
2. Solve for x (check your answer(s)): a.
3
5

9 x b.
3 x

5 x

4
3. Solve these proportions: a.
6
8
 x
21
 x b. x
5

45 x
 c.
9 x x y
 x
16 c. d.
4 x

3
 x x

8 x

3
4

2
3

4. If P is a point on AB , and AP
PB

5 , write (as numbers) AP
3 AB
and AB
PB
.
5. AB , BC , and CA are proportional to XY , YZ , and ZX . If AB
XY

3
5
, YZ
ZX

15.5
in , find the lengths BC and CA .

27.5
in ,
6) Given PA QB RC SD , AB

5 cm , BC

2 cm , QR

3 cm , and RS

4 cm .
Find PQ and CD .
P
A
Q
R
B
S
C
D

7) Suppose we are given AD bisects

A , AB

9 , AC

7 , and BC

8 . Find BD and DC .
B
A
D
C
8) The perimeter of a triangle is 25cm. The bisector of one angle divides the opposite side into segments 6 and 4 cm. long. Find the other sides of the triangle.

9) Given each of the following diagrams, find the values of the unknown length(s). Show sufficient work to illustrate total understanding of the relevant concepts. b. c. a.
5
X
X+3
3.5
4
8 x
5 d. y
8 x x y
3
15
7
3
4