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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

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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

Find the Value of ∠ADB

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Answer

To find the angle ∠ADB, we can use the fact that the angles in a triangle sum to 180°. Therefore:

∠ADB=180°−∠BDA−∠BCD∠ADB = 180° - ∠BDA - ∠BCD

Substituting the known values:

∠ADB=180°−65°−110°=5°∠ADB = 180° - 65° - 110° = 5°

Step 2

Determine the Value of ∠TAD

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Answer

Since AT and BT are tangents to the circle from points A and B, we know that:

∠TAD=∠ADB∠TAD = ∠ADB

Thus:

∠TAD=5°∠TAD = 5°

Step 3

Calculate the Full Angle ∠TAD

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Answer

To find ∠TAD, we also need to consider the relation:

∠TAD+∠ADB+∠BCD=180°∠TAD + ∠ADB + ∠BCD = 180°

Thus:

∠TAD=180°−∠ADB−∠BCD∠TAD = 180° - ∠ADB - ∠BCD

Now substituting the known angles:

∠TAD=180°−5°−110°=65°∠TAD = 180° - 5° - 110° = 65°

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