## Math 8 Chapter 2 Lesson 5: Area of a rhombus

## 1. Theoretical Summary

### 1.1. How to calculate the area of a quadrilateral with two perpendicular diagonals

The area of a quadrilateral with two perpendicular diagonals is half the product of their lengths.

\({S_{ABCD}}=\dfrac{1}{2} AC. BD\)

**Comment:** The area of a quadrilateral with two perpendicular diagonals is half the product of the two diagonals.

### 1.2. Area of rhombus

The area of a rhombus is half the product of the lengths of the two diagonals.

\(S=\dfrac{1}{2}{d_1}. {d_2}\)

**– Note:** A rhombus is also a special parallelogram, so we can use the formula for the area of a parallelogram to calculate the area of a rhombus.

## 2. Illustrated exercise

### 2.1. Exercise 1

Calculate the area of quadrilateral \(ABCD\) in \(AC, BD\), knowing \(AC ⊥ BD\) at \(H\) (h.\(145\))

**Solution guide**

\({S_{ABC}} = \dfrac{1}{2}BH.AC\)

\({S_{ADC}}=\dfrac{1}{2}DH.AC\)

\({S_{ABCD}} = {S_{ABC}} + {S_{ADC}}\)\( \,= \dfrac{1}{2}BH.AC + \dfrac{1}{2}DH .AC\)\(\, = \dfrac{1}{2}(BH + DH).AC = \dfrac{1}{2}BD.AC\)

### 2.2. Exercise 2

a) Draw a quadrilateral with two diagonals of length \(3.6cm; 6cm\) and those two diagonals are perpendicular to each other. How many such quadrilaterals can be drawn? Calculate the area of each quadrilateral just drawn?

b) Calculate the area of a square whose diagonal length is \(d\).

**Solution guide**

a) Students draw quadrilaterals that satisfy the conditions of the problem, such as the quadrilateral \(ABCD\) in the figure with:

\(AC = 6cm\)

\(BD = 3.6cm\)

\(AC \perp BD\)

Can draw as many quadrilaterals as required from the problem.

The area of the quadrilateral just drawn is:

\(S_{ABCD}= \dfrac{1}{2} AC. BD = \dfrac{1}{2}6. 3.6 = 10.8\) (\(cm^2\))

b) Area of a square whose diagonal length is \(d\)

The square has two equal and perpendicular diagonals, so the area is:

\(S = \dfrac{1}{2} dd = \dfrac{1}{2} d^2\)

### 2.3. Exercise 3

Draw a rectangle whose side is equal to the diagonal of a given rhombus and whose area is equal to that of the rhombus. From this, deduce how to calculate the area of the rhombus.

**Solution guide**

Given a rhombus \(ABCD\) whose diagonals intersect at \(I\). Infer that \(I\) is the midpoint of \(AC\) or \(IC=\dfrac{1}{2}\) \(AC\) (property)

Draw a rectangle with one side diagonal \(BD\), the other side equal \(IC\) ( \(IC=\dfrac{1}{2}\) \(AC\))

Then the area of rectangle \(BFED\) is equal to the area of rhombus \(ABCD\).

Indeed:

\({S_{BF{\rm{ED}}}} = BD.IC = B{\rm{D}}.\dfrac{1}{2}AC \)\(= \dfrac{1}{2 }B{\rm{D}}.AC = {S_{ABC{\rm{D}}}}.\)

Then deduce how to calculate the area of the rhombus: The area of the rhombus is half the product of the two diagonals.

## 3. Practice

### 3.1. Essay exercises

**Question 1: **Of the rhombuses of equal perimeter, find the rhombus with the largest area.

**Verse 2:** Find the area of a rhombus, given that its side is \(6.2\,cm\) and one of its angles has a measure of \(30°\)

**Question 3: **Let \(ABCD,\) know that \(AB = 5cm,\, AI = 3cm\) (\(I\) is the intersection of two diagonals). Calculate the area of that rhombus.

**Question 4: **

a) Draw a quadrilateral with two diagonals perpendicular to each other, knowing the lengths of the diagonals \(a\) and \(\dfrac{1}{2}a\). How many such shapes can be drawn?

b) How many rhombus can be drawn, given that the lengths of the two diagonals are \(a\) and \(\dfrac{1}{2}a\) ?

c) Calculate the area of the shapes just drawn.

### 3.2. Multiple choice exercises

**Question 1:** The two diagonals of a rhombus are 6cm and 8cm long. The side lengths of the rhombus are:

A. 6 cm

B. 5 cm

C. 3 cm

D. 4 cm

**Verse 2: **Given rhombus MNPQ. Knowing A, B, C, D are the midpoints of the sides NM, NP, PQ, QM, respectively. Calculate the ratio of the areas of quadrilateral ABCD and rhombus MNPQ

A. \(\frac{1}{2}\)

B. \(\frac{2}{3}\)

C. 2

D. \(\frac{1}{3}\)

**Question 3: **A rhombus has two diagonals of length 6cm and 8cm. Calculate the length of the altitude of the rhombus

A. 9.6 cm

B. 4.8 cm

C. 3.6 cm

D. 5.4 cm

**Question 4: **Let ABCD be a parallelogram with AB = BC = 10 cm and O the intersection of the two diagonals such that OA = 6 cm. Calculate the area of parallelogram ABCD

A. 96

B. 80

C. 72

D. 64

**Question 5:** Let ABCD be a rhombus whose area is 24cm2. The ratio of the lengths of the two diagonals is 3: 4. Find the lengths of the two diagonals of the rhombus

A. 9cm and 12cm

B. 12cm and 16cm

C. 6cm and 8cm

D. 3cm and 4cm

## 4. Conclusion

Through this lesson, you will learn some of the main topics as follows:

- Know the formula for calculating the area of a rhombus
- Know two ways to calculate the area of a rhombus, know how to calculate the area of a quadrilateral with two perpendicular diagonals.
- Draw a rhombus correctly and discover and prove the theorem about the area of a rhombus.

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