Rhombus – Area of rhombus

### 1.1. Rhombus

A rhombus ABCD has:

- Side AB is parallel to side DC.

Side AD is parallel to side BC.

**A rhombus has two pairs of parallel opposite sides and four equal sides.**

### 1.2. Area of rhombus

Let ABCD rhombus with AC = m, BD = n.

Cut triangle AOD and triangle COD and then combine with triangle ABC to get rectangle MNCA (see figure).

Based on the figure we have:

The area of rhombus ABCD is equal to the area of rectangle MNCA.

The area of rectangle MNCA is \(m \times \frac{n}{2}\). Which \(m \times \frac{n}{2} = \frac{{m \times n}}{2}\).

So the area of rhombus ABCD is \(\frac{{m \times n}}{2}\)..

** The area of a rhombus is the product of the lengths of the two diagonals divided by 2** (same unit of measure).

\(S=\frac{{m \times n}}{2}\).

(S is the area of the rhombus and m; n is the lengths of the two diagonals).

### 1.3. Solve the textbook exercises pages 140, 141

**Lesson 1: **In the pictures below

Which figure is a rhombus?

Which shape is a rectangle?

__Solution guide:__

- Observe the figure to determine the names of the given shapes.
- A rhombus has two pairs of parallel sides with four equal sides.

Figure 1, Figure 3 are rhombus.

Figure 2 is a rectangle.

*Say more:* Figure 4 is a parallelogram, figure 5 is a square trapezoid.

**Lesson 2: **In rhombus ABCD, AC and BD are two diagonals of the rhombus that intersect at point O.

a) Use an eke to check if two diagonals are perpendicular to each other.

b) Use a ruler with centimeter divisions to check whether the two diagonals intersect at the midpoint of each.

__Solution guide:__

- Use a ruler and a ruler to check according to the requirements of the problem.

a) Check will see two diagonals perpendicular to each other.

b) Two diagonals intersect at the midpoint of each line because when we check we see:

OA = OC = 3cm ; OB = OD = 2cm.

Comment : *A rhombus has two diagonals that are perpendicular to each other and intersect at the midpoint of each line*.

**Lesson 3: **Practice folding and cutting the paper (according to the picture) to make a rhombus.

__Solution guide:__

- Students observe and practice according to the picture.

Students look closely at the picture to understand the lesson and practice according to the picture.

### 1.4. Solve textbook exercises pages 142, 143

**Lesson 1: **Calculate the area of

a) A rhombus ABCD, knowing AC = 3cm, BD = 4cm.

b) Rhombus MNPQ, know MP = 7cm, NQ = 4cm.

__Solution guide:__

The area of a rhombus is equal to the product of the lengths of the two diagonals divided by 2 (same units).

\(S = \frac{{m \times n}}{2}\) or S = m × n : 2

(S is the area of the rhombus; m, n are the lengths of the two diagonals).

a) The area of rhombus ABCD is:

\(\frac{{3 \times 4}}{2} = \frac{{12}}{2} = 6\left( {c{m^2}} \right)\)

b) The area of rhombus MNPQ is:

\(\frac{{7 \times 4}}{2} = \frac{{24}}{2} = 14\left( {c{m^2}} \right)\)

**Lesson 2: **Calculate the area of a rhombus given

a) The lengths of the diagonals are 5dm and 20dm;

b) The lengths of the diagonals are 4m and 15dm.

__Solution guide:__

- If the lengths of the two diagonals are not in the same unit, then we convert them to the same unit of measure, then to calculate the area of the rhombus we take the product of the lengths of the two diagonals divided by 2.

a) The area of the rhombus is:

\(\frac{{5 \times 20}}{2} = \frac{{100}}{2} = 50\left( {d{m^2}} \right)\)

b) Change: 4m = 40dm

The area of the rhombus is:

\(\frac{{40 \times 15}}{2} = \frac{{600}}{2} = 300\left( {d{m^2}} \right)\)

**Lesson 3: **Correct write D, false write WILL

a) The area of the rhombus is equal to the area of the rectangle.

b) The area of the rhombus is \(\frac{1}{2}\) the area of the rectangle.

__Solution guide:__

- To calculate the area of a rhombus we take the product of the lengths of the two diagonals divided by 2 (same unit of measure).
- To calculate the area of a rectangle, multiply the length by the width.

The area of the rhombus is:

5 × 2 : 2 = 5 (cm^{2})

The area of the rectangle is:

5 × 2 = 10 (cm^{2})

We have : 10 : 5 = 2

So the area of the rectangle is 2 times the area of the rhombus, or the area of the rhombus equals \(\frac{1}{2}\) the area of the rectangle.

a) Write S in the blank.

b) Write D in the blank box.

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