The diagram below shows the circle k (not to scale) - Junior Cycle Mathematics - Question 12 - 2022
Question 12
The diagram below shows the circle k (not to scale).
The points A, B, and C lie on the circle.
[AB] is a diameter of the circle, and |AC| = 8 cm.
The area of the cir... show full transcript
Worked Solution & Example Answer:The diagram below shows the circle k (not to scale) - Junior Cycle Mathematics - Question 12 - 2022
Step 1
Find the Diameter of Circle k
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Answer
The area of the circle is given by the formula:
A=πr2.
Since the area is 25π cm², we have:
25π=πr2.
Dividing both sides by π gives:
r2=25,
which results in:
r=5extcm.
Thus, the diameter, which is twice the radius, is:
d=2r=10extcm.
Step 2
Identify Triangle ABC Properties
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Answer
In triangle ABC, since AB is the diameter, angle ACB is a right angle (90°) according to the inscribed angle theorem. Therefore, we can label the angles:
∣CBA∣=x
∣CAB∣=90°−x.
Step 3
|AC| = 8 cm
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Answer
According to the given information, |AC| = 8 cm. We can now set up the sine ratio to solve for angle CBA:
rac{|AC|}{|AB|} = rac{8}{10} = rac{4}{5},
leading to:
ext{sin} |CBA| = rac{4}{5}.
Step 4
Calculate Angle CBA
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Answer
To find angle CBA, take the inverse sine:
|CBA| = ext{sin}^{-1}igg( rac{4}{5}igg),
which approximately equals 53.13°.
Step 5
Determine Angle CAB
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Answer
Since angle ACB is 90°:
∣CAB∣=90°−∣CBA∣=90°−53.13°=36.87°.
Thus, the angles of triangle ABC are:
∣CBA∣≈53.13°
∣CAB∣≈36.87°
∣ACB∣=90°.
Step 6
Identify the Smallest Angle
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Answer
The smallest angle in triangle ABC is:
∣CAB∣≈36.87°.
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