\(\frac{\text{Small Side A}}{\text{Large Side A}} = \frac{\text{Small Side B}}{\text{Large Side B}}\)

Indirect Measurement Case

6ft

4ft shadow

\(x\)

16ft shadow

\(\frac{6}{x} = \frac{4}{16} \implies 4x = 96 \implies x = 24\)

FILE 08

Proportional Relationships

Triangle Proportionality Theorem

If a line is parallel to one side of a triangle and intersects the other two sides, then it divides the two sides proportionally.

If \(DE \parallel BC\), then:

\(\frac{AD}{DB} = \frac{AE}{EC}\)

A D E B C a b c d

Parallel Indicator

GEOMETRY INVESTIGATION UNIT AGENT RECORD #SIM-04-C PAGE 3 OF 3

Equation Skeleton

\(\frac{\text{Small Figure}}{\text{Large Figure}} = \frac{\text{Small Figure}}{\text{Large Figure}}\)

Indirect Measurement Case

6ft

4ft shadow

\(x\)

16ft shadow

\(\frac{6}{x} = \frac{4}{16} \implies 4x = 96 \implies x = 24\)

FILE 08

Proportional Relationships

Triangle Proportionality Theorem

If a line is parallel to one side of a triangle, it divides the other two sides proportionally.

If \(DE \parallel BC\), then:

\(\frac{AD}{DB} = \frac{AE}{EC}\)

A D E B C a b c d

Parallel Indicators

GEOMETRY INVESTIGATION UNIT AGENT RECORD #SIM-04-C PAGE 3 OF 3

\(\frac{\text{Known}_1}{\text{Target}_x} = \frac{\text{Ratio Part}_1}{\text{Ratio Part}_2}\)

10

x

Shadow Mapping

REF #SIM-26-P2 SUBJECT: TRIANGLE INVESTIGATION

Advanced Intel

Proportional Relations

Extended Theorems & Relationships

THEOREMS 08

Triangle Proportionality

Theorem: If a line is parallel to one side of a triangle and intersects the other two sides, it divides them proportionally.

If \(DE \parallel BC\), then: \(\frac{AD}{DB} = \frac{AE}{EC}\)

Parallel Evidence Required

THEOREMS 09

Parallel Line Proportions

Theorem: If three or more parallel lines intersect two transversals, then they cut the transversals proportionally.

\(\frac{AB}{BC} = \frac{DE}{EF}\)

THEOREMS 10

Similarity & Scaling

Proportional Perimeters & Areas

Perimeter Ratio

Matches the scale factor of sides.

\(k\)

Area Ratio

Matches the square of the scale factor.

\(k^2\)

Agent Note: If side ratio is \(2:3\), then perimeter ratio is \(2:3\) and area ratio is \(4:9\).

REF #SIM-26-P3 SUBJECT: ADVANCED PROPORTIONALITY

\(\frac{a}{b} = \frac{c}{d}\)

Chart 09

Parallel Lines Theorem

Theorem: 3+ parallel lines cut transversals into proportional segments.

Chart 10

Similarity & Scaling

The Relation of Dimensions

Perimeter Ratio

\(k\)

(Same as sides)

Area Ratio

\(k^2\)

(Sides Squared)

Field Example:

If side ratio is 1:3,

then area ratio is 1:9.

File Ref: 404-SIM-03 Investigation Phase III: Scaling

Triangle Proportionality

The Theorem: If a line parallel to one side of a triangle intersects the other two sides, it divides them proportionally.

\(\frac{\text{Part A}}{\text{Part B}} = \frac{\text{Part C}}{\text{Part D}}\)

Parallel Condition Required

Chart 09

Parallel Line Proportions

The Theorem: If three or more parallel lines intersect two transversals, then they divide the transversals proportionally.

Chart 10

Similarity & Scaling

Proportionality of Dimensions

Perimeter Ratio

\(k\)

(Matching Side Ratio)

Area Ratio

\(k^2\)

(Side Ratio Squared)

Proof Example:

If side ratio is 2:5,

then area ratio is 4:25.

File Ref: 404-SIM-P3 Investigation Phase III: Scaling Laws

The Theorem: If a line parallel to one side of a triangle intersects the other two sides, it divides them proportionally.

\(\frac{\text{Part A}}{\text{Part B}} = \frac{\text{Part C}}{\text{Part D}}\)

Parallel Condition Required

Chart 09

Parallel Line Proportions

The Theorem: If three or more parallel lines intersect two transversals, then they divide the transversals proportionally.

Chart 10

Similarity & Scaling

Proportionality of Dimensions

Perimeter Ratio

1

2

\(k\)

Area Ratio

1

\(k^2\)

Proof Example:

If side ratio is 2:5,

then area ratio is 4:25.

File Ref: 404-SIM-P3 Investigation Phase III: Scaling Laws

Extended Laws of Similarity

Chart 08

Triangle Proportionality

The Theorem: If a line parallel to one side of a triangle intersects the other two sides, it divides them proportionally.

\(\frac{\text{Part A}}{\text{Part B}} = \frac{\text{Part C}}{\text{Part D}}\)

Parallel Condition Required

Chart 09

Parallel Line Proportions

The Theorem: If three or more parallel lines intersect two transversals, then they divide the transversals proportionally.

Chart 10

Similarity & Scaling

Proportionality of Dimensions

Perimeter Ratio

1

2

\(k\)

Area Ratio

1

\(k^2\)

Proof Example:

If side ratio is 2:5,

then area ratio is 4:25.

File Ref: 404-SIM-P3 Investigation Phase III: Scaling Laws

Proportionality Theorems

Extended Laws of Similarity

Chart 08

Triangle Proportionality

The Theorem: If a line parallel to one side of a triangle intersects the other two sides, it divides them proportionally.

\(\frac{\text{Part A}}{\text{Part B}} = \frac{\text{Part C}}{\text{Part D}}\)

A B C D

Parallel Condition Required

Chart 09

Parallel Line Proportions

The Theorem: If three or more parallel lines intersect two transversals, then they divide the transversals proportionally.

A

C

B

D

\(\frac{A}{B} = \frac{C}{D}\)

Chart 10

Similarity & Scaling

Proportionality of Dimensions

Perimeter Ratio

1

2

\(k\)

Area Ratio

1

\(k^2\)

Proof Example:

If side ratio is 2:5,

then area ratio is 4:25.

File Ref: 404-SIM-P3 Investigation Phase III: Scaling Laws

Advanced Intel

Proportionality Theorems

Extended Geometric Laws

Chart 08

Triangle Proportionality

The Theorem: A line parallel to one side of a triangle intersects the other two sides, it divides them proportionally.

\(\frac{A}{B} = \frac{C}{D}\)

A B C D

Condition: Parallel Line (\(\parallel\))

Chart 09

Parallel Line Proportions

The Theorem: 3+ parallel lines cut transversals into proportional segments.

W

Y

X

Z

\(\frac{W}{X} = \frac{Y}{Z}\)

Chart 10

Similarity & Scaling

Proportionality of Dimensions

Perimeter Ratio

1

2

\(k\)

(Same as sides)

Area Ratio

1

\(k^2\)

(Sides squared)

Field Evidence:

If side ratio is 2:5,

then Area ratio is 4:25.

File Ref: 404-SIM-P3 Investigation: Scaling Laws

Advanced Intel

Proportionality Theorems

Extended Laws of Similarity

Chart 08

Triangle Proportionality

The Theorem: If a line is parallel to one side of a triangle, it divides the other two sides proportionally.

\(\frac{A}{B} = \frac{C}{D}\)

A B C D

Condition: Parallel Line (\(\parallel\))

Chart 09

Parallel Line Proportions

The Theorem: 3+ parallel lines cut transversals into proportional segments.

W

Y

X

Z

\(\frac{W}{X} = \frac{Y}{Z}\)

Chart 10

Similarity & Scaling

Proportionality of Dimensions

Perimeter Ratio

1

k

\(k\)

Linear Scale

Area Ratio

1

\(k^2\)

Squared Scale

Case Study:

If side ratio is 2:5 (\(k = 2.5\)),

then Area ratio is 4:25 (\(k^2 = 6.25\)).

File Ref: 404-SIM-P3 Investigation: Scaling Laws

Advanced Intel

Proportionality Theorems

Extended Laws of Similarity

Chart 08

Triangle Proportionality

The Theorem: If a line is parallel to one side of a triangle, it divides the other two sides proportionally.

\(\frac{A}{B} = \frac{C}{D}\)

A B C D

Condition: Parallel Line (\(\parallel\))

Chart 09

Parallel Line Proportions

The Theorem: 3+ parallel lines cut transversals into proportional segments.

W

Y

X

Z

\(\frac{W}{X} = \frac{Y}{Z}\)

Chart 10

Similarity Scaling Theorem

Proportionality of Dimensions

Perimeter Ratio

1

k

\(k\)

Linear Scale

Area Ratio

1

\(k^2\)

Squared Scale

Case Study:

If side ratio is 2:5 (\(k = 2.5\)),

then Area ratio is 4:25 (\(k^2 = 6.25\)).

File Ref: 404-SIM-P3 Investigation: Scaling Laws

x

Shadow B

\(\frac{6}{x} = \frac{\text{Shadow A}}{\text{Shadow B}}\)

REF: SIM-DOSSIER-P3 Investigation Phase III: Application

Intel Update

Extended Laws

Chart 08

Triangle Proportionality

Theorem: If a line is PARALLEL to one side of a triangle, it divides the other two sides proportionally.

\(\frac{A}{B} = \frac{C}{D}\)

A B C D

Symbol: \(\parallel\)

Chart 09

Parallel Line Proportions

Theorem: If three or more parallel lines intersect two transversals, they divide them proportionally.

W

Y

X

Z

\(\frac{W}{X} = \frac{Y}{Z}\)

Chart 10

The Scaling Laws

Perimeter Ratio

1

k

\(k\)

Linear Proportions

Area Ratio

1

\(k^2\)

Squared Proportions

Field Note:
If the Side Ratio is 2:5,
then the Area Ratio is 4:25.

REF: SIM-DOSSIER-P4 Investigation Phase IV: Laws

\(\frac{A}{B} = \frac{C}{D}\)

A B C D

REF: SIM-DOS-03 Investigation Phase III

Intel HQ

Extended Laws

Chart 09

Parallel Line Proportions

Theorem: 3+ parallel lines divide transversals proportionally.

W

Y

X

Z

\(\frac{W}{X} = \frac{Y}{Z}\)

Chart 10

Dimension Scaling Laws

Perimeter Ratio

1

k

\(k\)

Area Ratio

1

\(k^2\)

Case Brief:

Side Ratio 2:5 (\(k = 2.5\)) → Area Ratio 4:25 (\(k^2 = 6.25\))

REF: SIM-DOS-04 Investigation Concluded