CK-12 Geometry-Concepts

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

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James Sousa: Applications of Proportions

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James Sousa: Using Similar Triangles to Determine Unknown Values

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.