CK-12 Geometry Concepts

http://www.ck12.org Chapter 7. Similarity

Guidance

Aproportionis when two ratios are set equal to each other.

Cross-Multiplication Theorem:Leta,b,c,anddbe real numbers, withb 6 =0 andd 6 =0. Ifab=cd, thenad=bc.

The proof of the Cross-Multiplication Theorem is an algebraic proof. Recall that multiplying by^22 ,bb,ordd= 1
because it is the same number divided by itself(b÷b= 1 ).

Proof of the Cross-Multiplication Theorem:

a

b

=

c

d

Multiply the left side by

d

d

and the right side by

b

b

.

a

b

·

d

d

=

c

d

·

b

b

ad

bd

=

bc

bd

The denominators are the same, so the numerators are equal.

ad=bc

Think of the Cross-Multiplication Theorem as a shortcut. Without this theorem, you would have to go through all
of these steps every time to solve a proportion. The Cross-Multiplication Theorem has several sub-theorems that
follow from its proof. The formal term iscorollary.

Corollary #1:Ifa,b,c,anddare nonzero andab=dc, thenac=bd.

Corollary #2:Ifa,b,c,anddare nonzero andab=dc, thendb=ca.

Corollary #3:Ifa,b,c,anddare nonzero andab=dc, thenba=dc.

Corollary #4:Ifa,b,c,anddare nonzero andab=dc, thena+bb=c+dd.

Corollary #5:Ifa,b,c,anddare nonzero andab=dc, thena−bb=c−dd.

Example A

Solve the proportions.

a)^45 = 30 x

b)y+ 81 = 205

c)^65 =^2 xx−+ 24

To solve a proportion, you need tocross-multiply.

a)

b)

c)

Example B

Your parents have an architect’s drawing of their home. On the paper, the house’s dimensions are 36 in by 30 in. If
the shorter length of your parents’ house is actually 50 feet, what is the longer length?

Set up a proportion. If the shorter length is 50 feet, then it will line up with 30 in. It does not matter which numbers
you put in the numerators of the fractions, as long as they line up correctly.