The points A, B, C and D lie on a circle and the tangents at A and B meet at T, as shown in the diagram - HSC - SSCE Mathematics Extension 1 - Question 3 - 2017 - Paper 1
Question 3
The points A, B, C and D lie on a circle and the tangents at A and B meet at T, as shown in the diagram.
The angles BDA and BCD are 65° and 110° respectively.
What... show full transcript
Worked Solution & Example Answer:The points A, B, C and D lie on a circle and the tangents at A and B meet at T, as shown in the diagram - HSC - SSCE Mathematics Extension 1 - Question 3 - 2017 - Paper 1
Step 1
Calculate ∠ADB
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Answer
Since ∠BDA is 65°, we can find ∠ADB as follows:
∠ADB=180°−∠BDA=180°−65°=115°
Step 2
Calculate ∠ABC
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Answer
For the angle ∠BCD, we have:
∠ABC=∠BCD=110°
Step 3
Use the circle theorem
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Answer
In a circle, the angle formed by a tangent and a chord through the point of contact is equal to the angle in the alternate segment. Hence:
∠TAD=∠ABC
So, we have:
∠TAD=110°
Step 4
Use the property of angles
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Now summing the angles around point A:
∠TAD+∠ADB=180°
Substituting the values we know:
∠TAD+115°=180°
Therefore:
∠TAD=180°−115°=65°
Step 5
Conclusion
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Therefore, using all the calculations, the final value of ∠TAD is:
