Question

secondary one mathematics exam papers g3

1. stanley draws the following diagram shown below during his art class. it consists of a semi-circle inside a trapezium with a right-angled triangle at the bottom as shown in the figure.

if the area of trapezium \\(abce\\) is \\(144\text{ cm}^2\\), taking \\(\pi = \frac{22}{7}\\), find
(i) \\(ce\\),
(ii) the area of the shaded region.

Explanation:

🆕 New Concept Discovered: Area of Composite Figures
Finding areas by combining or subtracting basic shapes.

Step 1: Find the length of \( CE \)

The area of a trapezium is given by the formula:

$$ \text{Area} = \frac{1}{2} \times (a + b) \times h $$

For trapezium \( ABCE \):

• Parallel sides are \( AB = 18\text{ cm} \) and \( CE \).
• Height \( h = 9\text{ cm} \).
• Area is given as \( 144\text{ cm}^2 \).

Substitute these values into the formula to solve for \( CE \):

$$ 144 = \frac{1}{2} \times (18 + CE) \times 9 $$

$$ 144 = 4.5 \times (18 + CE) $$

$$ 18 + CE = \frac{144}{4.5} $$

$$ 18 + CE = 32 $$

$$ CE = 32 - 18 = 14\text{ cm} $$

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Step 2: Calculate the area of the unshaded semi-circle

The semi-circle sits on the line segment \( CE \), which means its diameter is equal to the length of \( CE \).

• Diameter \( d = 14\text{ cm} \)
• Radius \( r = \frac{14}{2} = 7\text{ cm} \)

Using \( \pi = \frac{22}{7} \), the area of the semi-circle is:

$$ \text{Area of semi-circle} = \frac{1}{2} \times \pi r^2 $$

$$ \text{Area of semi-circle} = \frac{1}{2} \times \frac{22}{7} \times 7^2 $$

$$ \text{Area of semi-circle} = \frac{1}{2} \times \frac{22}{7} \times 49 $$

$$ \text{Area of semi-circle} = 11 \times 7 = 77\text{ cm}^2 $$

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Step 3: Calculate the shaded region in the trapezium

The shaded area in the upper section is the area of trapezium \( ABCE \) minus the area of the semi-circle:

$$ \text{Upper Shaded Area} = 144 - 77 = 67\text{ cm}^2 $$

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Step 4: Calculate the shaded area of the right-angled triangle

The bottom section is a right-angled triangle \( ECD \) with:

• Base \( CE = 14\text{ cm} \)
• Height \( CD = 9\text{ cm} \)

The area of this triangle is:

$$ \text{Area of triangle } ECD = \frac{1}{2} \times \text{base} \times \text{height} $$

$$ \text{Area of triangle } ECD = \frac{1}{2} \times 14 \times 9 = 63\text{ cm}^2 $$

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Step 5: Calculate the total shaded area

Sum the shaded areas from both sections:

$$ \text{Total Shaded Area} = \text{Upper Shaded Area} + \text{Area of triangle } ECD $$

$$ \text{Total Shaded Area} = 67 + 63 = 130\text{ cm}^2 $$

Answer:

(i) \( 14\text{ cm} \)

(ii) \( 130\text{ cm}^2 \)