Two circles, c₁ and c₂, intersect at the points B and X, as shown - Leaving Cert Mathematics - Question Question 1 - 2014
Question Question 1
Two circles, c₁ and c₂, intersect at the points B and X, as shown.
The circle c₁ has diameter [AB].
The circle c₂ has diameter [BC].
The line CB is a tangent to c₁.
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Worked Solution & Example Answer:Two circles, c₁ and c₂, intersect at the points B and X, as shown - Leaving Cert Mathematics - Question Question 1 - 2014
Step 1
Join B to X
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Answer
Since [AB] is a diameter of c₁, angle |AXB| is inscribed in a semicircle. Therefore, by the inscribed angle theorem, it follows that:
∣AXB∣=90°.
Step 2
Consider circle c₂
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Answer
Since [BC] is a diameter of c₂, angle |XCB| is also inscribed in a semicircle. Consequently, we have:
∣XCB∣=90°.
Thus, X forms right angles with both diameters.
Step 3
Analyze angles A, X, and C
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Answer
The angles generated imply that the sum of angles |AXB| and |XCB| leads to:
∣AXC∣=∣AXB∣+∣XCB∣=90°+90°=180°.
This means points A, X, and C are collinear, proving that X lies on the line AC.
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