Plane Shapes

Grade 12 Math - Properties of Plane Shapes and Scale Drawing

Lesson Objectives

Identify and apply the properties of parallelograms, rhombuses, and kites
Differentiate between the properties of each quadrilateral
Use scale drawing to represent and solve geometrical problems

Lesson Introduction

Plane shapes such as parallelograms, rhombuses, and kites have unique geometrical properties that define their structure. These properties help in solving complex problems involving angles, sides, and diagonals. Additionally, scale drawing is a method used to represent real objects or diagrams accurately on paper, maintaining proportional relationships.

Core Lesson Content

Properties of Parallelograms

Opposite sides are equal and parallel
Opposite angles are equal
Diagonals bisect each other

Properties of Rhombuses

All sides are equal in length
Opposite angles are equal
Diagonals bisect each other at right angles
Diagonals bisect angles

Properties of Kites

Two pairs of adjacent sides are equal
One pair of opposite angles is equal
One diagonal bisects the other at right angles

Scale Drawing

Used to represent shapes accurately
Common in construction and technical drawing
Use a fixed ratio (e.g., 1:100 means 1 unit represents 100 units)

Worked Examples

Example 1: In a parallelogram, one angle is 70°. Find the other three angles.
Solution: Opposite angle = 70°, adjacent angle = 110°, opposite = 110°

Example 2: A rhombus has one angle of 60°. Find the other three angles.
Solution: Opposite = 60°, adjacent = 120°, opposite = 120°

Example 3: In a kite, two adjacent sides are 5 cm and 5 cm. Find the perimeter if the other two sides are 7 cm each.
Solution:

P = 5 + 5 + 7 + 7 = 24\, \text{cm}

Example 4: Diagonals of a rhombus are 10 cm and 24 cm. Find the side length.
Solution: Use Pythagoras:

s = \sqrt{(10/2)^2 + (24/2)^2} = \sqrt{25 + 144} = \sqrt{169} = 13\, \text{cm}

Example 5: Draw a parallelogram with adjacent sides 6 cm and 4 cm, and an angle of 60°. Use scale 1:1.
Solution: Draw side 6 cm, then angle 60°, draw side 4 cm, complete parallelogram.

Example 6: A scale drawing is made with a ratio 1:50. If a wall is 6 m, what is its length on the drawing?
Solution:

6\, \text{m} = 600\, \text{cm}, \quad 600/50 = 12\, \text{cm}

Example 7: One diagonal of a kite is 12 cm and the other is 16 cm. Find the area.
Solution:

A = \frac{1}{2} \times 12 \times 16 = 96\, \text{cm}^2

Example 8: Opposite sides of a quadrilateral are equal and one pair is parallel. Is it a parallelogram?
Solution: No. Both pairs must be parallel for it to be a parallelogram.

Example 9: A rhombus has side 10 cm and one angle 60°. Find area using

A = a^2 \sin(\theta)

.
Solution:

A = 10^2 \sin 60^\circ = 100 \times \frac{\sqrt{3}}{2} \approx 86.6\, \text{cm}^2

Example 10: In a scale drawing of a kite, actual diagonal is 30 cm, scale is 1:5. What is the drawn length?
Solution:

30\, \text{cm} / 5 = 6\, \text{cm}

Exercises

Find all angles in a parallelogram if one angle is 75°
[NECO] A kite has one pair of equal angles. Prove this using geometric reasoning [Past Question]
Draw a parallelogram using scale 1:2 with actual sides 8 cm and 6 cm
[WAEC] Draw a rhombus with sides 5 cm and one angle 45° [Past Question]
A rhombus has a diagonal of 20 cm and angle of 60°. Find the other diagonal
Calculate the area of a kite with diagonals 14 cm and 10 cm
Using scale 1:100, draw a building side 20 m long. What length is on paper?
[WAEC] A parallelogram has base 10 cm and height 6 cm. Find area using appropriate formula [Past Question]
Find the length of diagonals in a rhombus if each side is 10 cm and one diagonal is 12 cm
In a kite, sides are 6 cm and 6 cm, and the other two are 9 cm and 9 cm. Find perimeter

Conclusion/Recap

Understanding the properties of parallelograms, rhombuses, and kites allows you to solve complex geometric problems. Scale drawing connects theoretical geometry to practical applications such as architecture, design, and map reading.