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
Determine ∠ABC
96%
114 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
Since the angles at A and B are formed by tangents to a circle, we can use the property that the angle between a tangent and a chord is equal to the angle in the alternate segment. Thus,
∠ABC = ∠BDA = 65°.
Step 2
Calculate ∠ABD
99%
104 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
Next, we observe that the angles of triangle ABD sum up to 180°. Therefore, we have:
∠ABD+∠BDA+∠ABD=180°
Substituting the known values:
∠ABD+65°+110°=180°
This gives:
∠ABD=180°−175°=5°
Step 3
Use the tangent property to find ∠TAD
96%
101 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
Using the previously calculated angle ∠ABD, we use the fact that:
∠TAD=∠ABD+∠ABC
Substituting the known angles:
∠TAD=5°+65°=70°
Given that angle ∠TAD can also be expressed from the sum of angles around point T, we add:
∠TAD=180°−(65°+110°)=5°
This correlates back to our earlier calculations confirming:
