Exercise 9.4
Exercise 9.4 Solutions
Polygon angle theorems, area applications, tessellations, and practical scaling/tiling word problems.
Question 1
Polygon Angles
Solve the following polygon angle problems:
(i) What is the sum of the interior angles of a decagon (10-sided polygon)?
(ii) Calculate the measure of each interior angle of a regular hexagon.
(iii) What is each exterior angle of a regular pentagon?
(iv) If the sum of the interior angles of a polygon is 1260°, how many sides does the polygon have?
(i) What is the sum of the interior angles of a decagon (10-sided polygon)?
(ii) Calculate the measure of each interior angle of a regular hexagon.
(iii) What is each exterior angle of a regular pentagon?
(iv) If the sum of the interior angles of a polygon is 1260°, how many sides does the polygon have?
Solution Step-by-Step
(i) Sum of interior angles of decagon
The formula for the sum of the interior angles of an n-sided polygon is:
Sum
=
(n - 2) × 180°
For a decagon, n = 10:
Sum
=
(10 - 2) × 180° = 8 × 180° = 1440°
(ii) Measure of each interior angle of a regular hexagon
The formula for each interior angle of a regular n-sided polygon is:
Angle
=
(n - 2) × 180°n
For a hexagon, n = 6:
Angle
=
(6 - 2) × 180°6 = 4 × 180°6 = 4 × 30° = 120°
(iii) Each exterior angle of a regular pentagon
The formula for each exterior angle of a regular n-sided polygon is:
Exterior Angle
=
360°n
For a pentagon, n = 5:
Exterior Angle
=
360°5 = 72°
(iv) Number of sides for Sum = 1260°
Using the sum formula:
∴ The polygon has 9 sides (nonagon).
(n - 2) × 180°
=
1260°
n - 2
=
1260°180° = 7
n
=
7 + 2 = 9
Question 2
Parallelogram Area
In a parallelogram ABCD, AB = 10 cm, AD = 6 cm and ∠BAD = 45°. Calculate the area of ABCD.
Solution Step-by-Step
The formula for the area of a parallelogram is:
Area
=
base × height = AB × DE
where DE is the altitude perpendicular to base AB.
In right-angled triangle ΔAED (with right angle at E):
sin(45°)
=
OppositeHypotenuse = DEAD
1√2
=
h6 ⇒ h = 6√2 = 3√2 cm
Now, calculate the area of the parallelogram:
Area
=
10 × 3√2 = 30√2 cm² ≈ 42.43 cm²
∴ The area of the parallelogram is approximately 42.43 cm².
Question 3
Parallelogram Angles
In a parallelogram ABCD if ∠DAB = 70°, find the measures of all other angles in the parallelogram.
Solution Step-by-Step
In a parallelogram, opposite angles are equal, and consecutive angles sum to 180°:
∠C
=
∠A = 70° [Opposite angles]
∠A + ∠B
=
180° [Consecutive angles]
70° + ∠B
=
180° ⇒ ∠B = 180° - 70° = 110°
∠D
=
∠B = 110° [Opposite angles]
The measures of the angles are: ∠A = 70°, ∠B = 110°, ∠C = 70°, ∠D = 110°.
Question 4
Tessellating Square
A shape is created by cutting a square in half diagonally and then attaching a right-angled triangle to the hypotenuse of each half. Explain why this shape can tessellate and calculate the interior angles of the new shape.
Solution Step-by-Step
Tessellation explanation:
1. Cutting a square diagonally produces two right-angled isosceles triangles with angles of 45°, 45°, and 90°.
2. When we attach a right-angled triangle onto the hypotenuse of each half, the resulting shape is another square.
3. Since squares have interior angles of 90°, they can tessellate perfectly because four squares meet at a vertex:
1. Cutting a square diagonally produces two right-angled isosceles triangles with angles of 45°, 45°, and 90°.
2. When we attach a right-angled triangle onto the hypotenuse of each half, the resulting shape is another square.
3. Since squares have interior angles of 90°, they can tessellate perfectly because four squares meet at a vertex:
Vertex angle sum
=
4 × 90° = 360°
Since the sum at the meeting point is exactly 360°, they tile the plane with no gaps or overlaps.
Interior angle calculation:
Each corner of the constructed shape sums up to exactly 90°, making it a square.
Each corner of the constructed shape sums up to exactly 90°, making it a square.
Question 5
Triangle Reflections
A tessellation is created by repeatedly reflecting a basic shape. The basic shape is a right-angled triangle with sides of length 3, 4, and 5 units. Find: The minimum number of reflections needed to create a tessellation that covers a square with an area of 3600 square units.
Solution Step-by-Step
First, calculate the area of the basic right-angled triangle:
Areatriangle
=
12 × base × height = 12 × 3 × 4 = 6 square units
To cover a total area of 3600 square units, find the number of triangular tiles required:
Number of reflections
=
Total AreaArea of Triangle = 36006 = 600
Thus, a minimum of 600 reflections (or triangle tiles) are needed.
Question 6
Hexagon Tessellation
A tessellation is created using regular hexagons. Each hexagon has a side length of 5 cm. Find the total area of the tessellation if it consists of 25 hexagons, and find the total perimeter of the outer edge of the tessellation, assuming it's a perfect hexagon.
Solution Step-by-Step
1. Calculate the total area:
A regular hexagon consists of 6 equilateral triangles of side s = 5 cm.
A regular hexagon consists of 6 equilateral triangles of side s = 5 cm.
Area of 1 hexagon
=
6 × √34 × s2 = 6 × √34 × 25 = 37.5√3 cm² ≈ 64.952 cm²
Total area (25 hexagons)
=
25 × 64.952 ≈ 1623.8 cm²
2. Calculate the perimeter:
Assuming the combined tessellation itself forms a large regular hexagon:
Assuming the combined tessellation itself forms a large regular hexagon:
Arealarge
=
3√32 × x2 = 1623.8
where x is the side length of the large hexagon.
x2
=
1623.8 × 23√3 ≈ 625
x
=
√625 = 25 cm
Perimeter
=
6 × x = 6 × 25 = 150 cm
Thus, the total area is 1623.8 cm² and the outer perimeter is 150 cm.
Question 7
Floor Tiling
A rectangular floor is 12 m by 15 m. How many square tiles, each 1 m by 1 m, are needed to cover the floor?
Solution Step-by-Step
Find the area of the rectangular floor:
Areafloor
=
length × width = 12 × 15 = 180 m²
Find the area of one tile:
Areatile
=
1 × 1 = 1 m²
Find the number of tiles:
Number of tiles
=
1801 = 180
∴ A total of 180 tiles are needed.
Question 8
Wall Paint Gallons
A rectangular wall is 10 m tall and 120 m wide. How many gallons of paint are needed to cover the wall, if one gallon covers 35 m²?
Solution Step-by-Step
Calculate the wall area:
Areawall
=
10 × 120 = 1200 m²
Compute the gallons of paint needed:
Gallons
=
120035 ≈ 34.29 gallons
Since we cannot buy a fraction of a gallon, we round up:
∴ 35 gallons of paint are required.
Question 9
Wall Paint Liters
A rectangular wall has a length of 10 m and a width of 4 meters. If 1 litre of paint covers 7 m², how many liters of paint are needed to cover the wall?
Solution Step-by-Step
Calculate the wall area:
Areawall
=
10 × 4 = 40 m²
Compute the liters of paint needed:
Liters
=
407 ≈ 5.71 liters
Rounding up to the nearest whole liter:
∴ 6 liters of paint are required.
Question 10
Trapezoid Window
A window has a trapezoidal shape with parallel sides of 3 m and 1.5 m and a height of 2 m. Find the area of the window.
Solution Step-by-Step
The area of a trapezoid is given by the formula:
Area
=
a + b2 × h
where a = 3 m, b = 1.5 m are parallel sides, and h = 2 m.
Substitute the values:
Area
=
3 + 1.52 × 2
=
4.5 × 1 = 4.5 m²
∴ The area of the window is 4.5 m².
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