The circle c is shown in the diagram below (not to scale) - Leaving Cert Mathematics - Question 6 - 2022
Question 6
The circle c is shown in the diagram below (not to scale). Its centre is at the point O. The points A, B, and D lie on the circle, and [AB] is a diameter of the circ... show full transcript
Worked Solution & Example Answer:The circle c is shown in the diagram below (not to scale) - Leaving Cert Mathematics - Question 6 - 2022
Step 1
|∠ADB|, the size of the total angle at the point D.
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Answer
The angle |∠ADB| formed by the arc is subtended by the diameter, hence it measures 90°.
Step 2
|∠AOD| = 130°. Work out the size of the angle marked X in the diagram.
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Answer
Since |∠AOD| = 130°, the inscribed angle |∠AXB|, directly related to angle |∠AOD| where the arc subtended is angle AXB, is given by the formula:
X=21∣∠AOD∣=21(130°)=65°.
Step 3
The radius of the circle is 18 cm. Find the length of the arc AD.
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Answer
The arc length can be calculated using the formula:
Arc Length=360θ×2πr,
where θ is the angle at the center for arc AD, which is 130° and r is the radius.
Substituting the values:
Arc Length=360130×2π×18=3613×36π=13π cm.
Step 4
Statement A: If two triangles are similar, then they must be congruent.
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This statement is: false.
Reason: Similar triangles have the same shape but may differ in size; therefore, they are not necessarily congruent.
Step 5
Statement B: If two triangles are congruent, then they must be similar.
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Answer
This statement is: true.
Reason: Congruent triangles are indeed similar, as they have both the same shape and size, including the corresponding angles.
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