10.1 Give a reason why $ar{T}_S = ar{A}_2 = x$ - NSC Mathematics - Question 10 - 2017 - Paper 2
Question 10
10.1 Give a reason why $ar{T}_S = ar{A}_2 = x$.
10.2 Prove that:
10.2.1 $AB \parallel ST$
10.2.2 $ar{T}_4 = \bar{A}_1$
10.2.3 RTAP is a cyclic quadrilateral... show full transcript
Worked Solution & Example Answer:10.1 Give a reason why $ar{T}_S = ar{A}_2 = x$ - NSC Mathematics - Question 10 - 2017 - Paper 2
Step 1
Give a reason why $ar{T}_S = ar{A}_2 = x$.
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Answer
By the tangent-chord theorem, the angles formed by the tangent and the line touching the circle are equal, meaning that the angle ar{T}_S at point S is equal to the angle ar{A}_2 at point A. Since both angles are also equal to the value of x by definition in the problem, we conclude that ar{T}_S = ar{A}_2 = x.
Step 2
Prove that: 10.2.1 $AB \parallel ST$
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Answer
In triangle BRS, angles BAS and RTS are equal because they subtend the same arc BR. Therefore, by the Alternate Interior Angles Theorem, lines AB and ST must be parallel.
Step 3
Prove that: 10.2.2 $ar{T}_4 = \bar{A}_1$
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Answer
Given that B2=x, and since ar{T}_4 is thus an angle that subtends the same arc as ar{A}_1, it follows from the tangent-chord theorem that ar{T}_4 = ar{A}_1.
Step 4
Prove that: 10.2.3 RTAP is a cyclic quadrilateral.
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Answer
To prove that RTAP is a cyclic quadrilateral, we need to show that the opposite angles are supplementary. Since angles ar{A}_1 and ar{T}_4 add up to 180 degrees, we confirm that RTAP fulfills the cyclic quadrilateral condition.
