Chapter 4 · Plane Geometry

Lesson 4.5 — Complete Figures and Preserve Area Differences

第4讲 · 补全图形与面积差不变

An awkward pair of regions can become two familiar complete figures after the same area is added to both. The figures grow, but the gap between their areas does not change.

Add the same area to both sides—the difference stays fixed.

Grade 5 enrichment 50–65 minutes Layered SVG diagrams Autosaves in this browser

Mission 1

See an area difference as a fixed gap

Not completed

Suppose Region A is larger than Region B. If both regions receive the same added area, both totals grow by the same amount. Their difference stays unchanged.

Region A

Region B

Before and after the addition, A is 12 square units larger than B.

Smaller original area24

Original difference12

Area added to both18

Difference rule

A − B = k

(A + C) − (B + C) = k

1. For the current sliders, what is the difference after the addition?

square units

2. Which statement is true?

3. If the same area is added, can the larger region become the smaller one?

Mission 2

Complete two partial regions with one common area

Not completed

In a geometry diagram, the two awkward regions often touch the same unshaded piece. Add that common piece to each region. The added piece must not overlap the interior of either original region, so no area is counted twice. Each side may become a familiar whole—such as a rectangle, parallelogram, or triangle.

Original A

Equal added area

Original B

18 + 23 = 41

↔

11 + 23 = 34

41 − 34 = 7, exactly the same as 18 − 11 = 7.

1. What is the completed area on the A side?

square units

2. What is the completed area on the B side?

square units

3. What is the completed-area difference?

square units

Mission 3

Use the completion checklist

Not completed

To preserve an area difference, add equal amounts to both original areas. When joining pictured pieces, also check that they do not overlap within either completed figure. In diagram they are usually the same common region. Also preserve the direction of the statement: “A is 6 larger than B” remains “completed A is 6 larger than completed B.”

Checklist: identify the original two regions → identify equal added area → name the two completed wholes → keep the difference and its direction.

Case A: Add an area of 9 to each region.

Does this preserve the original area difference?

Case B: Add 9 only to the larger region.

Does this preserve the original area difference?

Case C: Add two differently shaped pieces, each with area 7.

Does this preserve the original area difference?

Case D: A is 10 larger than B. After equal additions, what remains true?

Mission 4

Build the chapter’s parallelogram–triangle diagram

Not completed

Chapter 4 · Example 6 A parallelogram and a right triangle overlap. The striped part of the parallelogram is 10 cm² larger than triangle FEG. The shared trapezoid BCFG completes both regions.

Parallelogram remainder

Common BCFG

Triangle FEG

Given: shaded remainder − Area(FEG) = 10

Add BCFG to both: (shaded + BCFG) − (FEG + BCFG) = 10

Completed figures: Area(ABCD) − Area(BCE) = 10

1. Which region is added to both original regions?

2. The shaded remainder plus BCFG becomes:

3. Triangle FEG plus BCFG becomes:

4. Which completed whole is larger, and by how much?

Mission 5

Use the completed figures to find CF

Not completed

Once the awkward regions are completed, the arithmetic is short. In Mission 4, right triangle BCE has perpendicular sides BC = 10 cm and CE = 8 cm. The parallelogram uses the same base BC; CF is its perpendicular height because F lies on AD and CE is perpendicular to BC.

Right triangle BCE10 × 8 ÷ 2 = 40 cm²

Transfer the differenceArea(ABCD) = 40 + 10 = 50 cm²

Work backward from parallelogram areaCF × 10 = 50

Missing heightCF = 5 cm

1. Area of right triangle BCE:

cm²

2. Area of parallelogram ABCD:

cm²

3. Length CF:

The parallelogram remains 10 cm² larger after completion.

Mission 6

Complete an external triangle into a rectangle problem

Not completed

Exercise 3 ABCD is a rectangle. E lies beyond D on the line CD, and EB meets AD at F. BC = 10 units and EC = 6 units. Triangle EDF is 5 square units smaller than triangle FAB. Add the common quadrilateral DCBF to both triangles.

Original relationArea(EDF) = Area(FAB) − 5

Add DCBF to bothArea(ECB) = Area(ABCD) − 5

Known complete triangle10 × 6 ÷ 2 = 30

Rectangle30 + 5 = 35

1. Area of right triangle ECB:

square units

2. Which completed whole is larger?

3. Area of rectangle ABCD:

square units

Mission 7

Preserve the difference, then work backward to ED

Not completed

Exercise 13 Rectangle ABCD overlaps triangle EBC. E lies beyond D on the line CD, and EB meets AD at F. Triangle EFD is 6 cm² larger than triangle ABF. The rectangle has BC = 6 cm and CD = 4 cm. Find ED.

Rectangle area6 × 4 = 24 cm²

Transfer the +6 differenceArea(EBC) = 24 + 6 = 30 cm²

Recover the full height EC6 × EC ÷ 2 = 30, so EC = 10 cm

Remove CDED = 10 − 4 = 6 cm

1. Rectangle area:

cm²

2. Area of complete triangle EBC:

cm²

3. Full height EC:

4. Length ED:

Mission 8

Complete a larger triangle, then subtract

Not completed

Classroom Practice 4 Square ABCD has side 12 cm. Point E divides DC so that DE is twice EC. Line AEF meets the extension of BC at F. Find the area of triangle DEF and the length CF. First complete triangle ADF; then remove triangle ADE.

Split side DC in the ratio 2 : 1DE = 8 cm, EC = 4 cm

Complete triangle ADFBase AD is 12 cm. F lies on the line BC, parallel to AD, so its perpendicular distance from AD is also 12 cm.12 × 12 ÷ 2 = 72 cm²

Remove triangle ADE72 − (12 × 8 ÷ 2) = 24 cm²

Use triangle DEF backwardUse DE = 8 cm as the base. CF is perpendicular to the line DE, so CF is its matching height.8 × CF ÷ 2 = 24, so CF = 6 cm

1. Length DE:

2. Area of complete triangle ADF:

cm²

3. Area of triangle DEF:

cm²

4. Length CF:

Mission 9

Area-difference workshop

Not completed

Use equal additions where needed, then calculate from the named mission’s measurements. Keep that mission’s units. Reach at least 6 out of 8.

1. Areas A and B satisfy A − B = 9 square units. Add 14 square units to each. Find completed A minus completed B.

2. A triangle has area 36 cm². A parallelogram has area 12 cm² larger and base 12 cm. Find its perpendicular height in cm.

3. In Missions 4–5’s parallelogram–triangle example, CF = ?

4. In Mission 6, rectangle ABCD area = ?

5. In Mission 7, ED = ?

6. In Mission 8’s square-extension example, area of triangle DEF = ?

7. In Mission 8’s square-extension example, CF = ?

8. A rectangle has area 42. After completing two regions, a triangle is 8 larger than the rectangle. Triangle area?

Hints

1: equal additions do not change the gap. 2: first find parallelogram area. 3–7: revisit the named mission diagrams. 8: keep the direction “triangle is larger.”

Worked workshop solutions — open after attempting

Find the first step that differs from your own work, then try the problem again.

Adding equal areas preserves the difference 9.
Parallelogram area = 36 + 12 = 48; height = 48 ÷ 12 = 4.
Triangle area = 10 × 8 ÷ 2 = 40. Add 10, then divide by base 10: CF = 5.
The completed triangle has area 10 × 6 ÷ 2 = 30. The rectangle is 5 larger: 35.
Rectangle area = 24. Triangle area = 30, so EC = 2 × 30 ÷ 6 = 10 and ED = 10 − 4 = 6.
DE = 8. Subtract ADE from ADF: 72 − 48 = 24.
Use DEF: 24 = ½ × 8 × CF, so CF = 6.
Keep the direction: triangle area = 42 + 8 = 50.

Mission 10

Exit ticket

Not completed

Try all five questions using the area relationships, then check your answers.

1. Which rule is correct?

2. In Missions 4–5’s parallelogram–triangle example, what is CF?

3. In Mission 6, what is the rectangle area?

square units

4. In Mission 7, what is ED?

5. If P is 7 square units larger than Q, what remains true after equal areas are added?

Optional reflection — not automatically graded: What complete figure would you try to create first in a new overlap problem?

Area-Difference Completer

You have completed this lesson.

You can identify a common region, complete awkward pieces into familiar figures, preserve the direction and size of an area difference, and work backward to a missing length.

Lesson 4.5 · Grade 5 Math Explorer

Teaching notes

The original states that parallelogram ABCD and right triangle BCE share trapezoid BCFG.Adding that trapezoid to the two original regions transfers the 10 cm² difference to the complete parallelogram and complete right triangle.With BC = 10 and CE = 8, the triangle area is 40 cm², the parallelogram area is 50 cm², and CF = 5 cm.

The numerical bar model, validity cases, feedback, workshop, and exit ticket are added instructional scaffolds.Diagrams are explanatory and are not scale drawings.