The diagram shows a circle with centre \(O\) and radius \(9\) cm. The line \(TA\) is a tangent to the circle at \(A\) and \(OT=22\) cm. (a) Calculate angle ...
Question 1 Report
The diagram shows a circle with centre \(O\) and radius \(9\) cm. The line \(TA\) is a tangent to the circle at \(A\) and \(OT=22\) cm.
(a) Calculate angle \(OTA\), giving your answer correct to \(1\) decimal place. [2]
(b) Find angle \(AOT\). [1]
Answer Details
\(TA\) is a tangent at \(A\), so the radius \(OA\) meets it at a right angle: angle \(OAT=90^\circ\). Triangle \(OAT\) is right-angled at \(A\), with \(OA=9\) cm and hypotenuse \(OT=22\) cm.
(a) Relative to angle \(OTA\), the radius \(OA\) is the opposite side and \(OT\) is the hypotenuse, so use the sine ratio:
\[ \sin(\text{angle } OTA)=\frac{9}{22}=0.40909\ldots \] [M1]Take the inverse sine:
\[ \text{angle } OTA=\sin^{-1}(0.40909\ldots)=24.147\ldots^\circ \]Correct to \(1\) decimal place, angle \(OTA=24.1^\circ\) [A1].
(b) The angles of triangle \(OAT\) sum to \(180^\circ\), and one of them is the right angle at \(A\), so the two remaining angles are complementary:
\[ \text{angle } AOT=90-24.147\ldots=65.85\ldots \]which is \(65.9^\circ\) [B1]. A candidate whose part (a) differs is credited for correctly subtracting their own value from \(90^\circ\).
Note that part (b) needs no new trigonometry: recognising the complementary pair saves time and avoids compounding rounding errors. Use the unrounded \(24.147\ldots^\circ\) in the subtraction, since starting from the rounded \(24.1^\circ\) gives \(65.9^\circ\) here but can shift the last digit in other questions.
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