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Here is a shaded shape ABCD - Edexcel - GCSE Maths - Question 17 - 2018 - Paper 3
Question 17
Here is a shaded shape ABCD. The shape is made from a triangle and a sector of a circle, centre O and radius 6 cm. OCD is a straight line. AD = 14 cm Angle AOD = 1... show full transcript
Worked Solution & Example Answer:Here is a shaded shape ABCD - Edexcel - GCSE Maths - Question 17 - 2018 - Paper 3
Step 1
Find the length of arc CD
Answer
To find the length of arc CD, we can use the formula for the arc length:
where r is the radius (6 cm) and \theta is the angle in radians. First, we need to convert the angle AOD (140°) to radians.
[ \theta = \frac{140 \times \pi}{180} = \frac{7\pi}{9} \text{ radians} ]
Now, we can calculate the length of arc CD:
[ L = 6 \times \frac{7\pi}{9} \approx 14.72 \text{ cm} ]
Step 2
Find the length OD
Answer
Since OCD is a straight line, we can find the triangle OAD using the sine rule. To find OD:
Using the triangle, we calculate:
[ \angle AOD + \angle OAD = 140° + 24° = 164° ]
Let’s find side OD using:
[ \frac{OD}{\sin(24°)} = \frac{14}{\sin(164°)} ]
Calculating the values:
[ OD = \frac{14 \cdot \sin(24°)}{\sin(164°)} \approx 5.67 \text{ cm} ]
Step 3
Calculate the perimeter of the shape
Answer
The perimeter of the shape can be calculated by summing the lengths AD, the length of arc CD, and the length OD:
[ P = AD + CD + OD ]
Substituting the values:
[ P = 14 + 14.72 + 5.67 \approx 34.39 \text{ cm} ]
Thus, rounding to 3 significant figures, we get:
[ P \approx 34.4 \text{ cm} ]