Proving angle relationships

7 \includegraphics[max width=\textwidth, alt={}, center]{10b2ec29-adca-4313-ae24-bab8b2d9f8a4-08_556_751_255_696} In the diagram the lengths of \(A B\) and \(A C\) are both 15 cm . The point \(P\) is the foot of the perpendicular from \(C\) to \(A B\). The length \(C P = 9 \mathrm {~cm}\). An arc of a circle with centre \(B\) passes through \(C\) and meets \(A B\) at \(Q\).

1. Show that angle \(A B C = 1.25\) radians, correct to 3 significant figures.
2. Calculate the area of the shaded region which is bounded by the \(\operatorname { arc } C Q\) and the lines \(C P\) and \(P Q\).

7 \includegraphics[max width=\textwidth, alt={}, center]{9362eb16-88c9-4279-97aa-907b4916b965-3_469_673_1720_737} The diagram shows triangle \(A B C\), with \(A B = 10 \mathrm {~cm} , B C = 13 \mathrm {~cm}\) and \(C A = 14 \mathrm {~cm} . E\) and \(F\) are points on \(A B\) and \(A C\) respectively such that \(A E = A F = 4 \mathrm {~cm}\). The sector \(A E F\) of a circle with centre \(A\) is removed to leave the shaded region \(E B C F\).

1. Show that angle \(C A B\) is 1.10 radians, correct to 3 significant figures.
2. Find the perimeter of the shaded region \(E B C F\).
3. Find the area of the shaded region \(E B C F\).

\includegraphics{figure_6} The diagram shows a sector \(OAB\) of a circle with centre \(O\). Angle \(AOB = \theta\) radians and \(OP = AP = x\).

1. Show that the arc length \(AB\) is \(2x\theta \cos \theta\). [2]
2. Find the area of the shaded region \(APB\) in terms of \(x\) and \(\theta\). [4]

\includegraphics{figure_2} The diagram shows a triangle \(AOB\) in which \(OA\) is 12 cm, \(OB\) is 5 cm and angle \(AOB\) is a right angle. Point \(P\) lies on \(AB\) and \(OP\) is an arc of a circle with centre \(A\). Point \(Q\) lies on \(AB\) and \(OQ\) is an arc of a circle with centre \(B\).

1. Show that angle \(BAO\) is 0.3948 radians, correct to 4 decimal places. [1]
2. Calculate the area of the shaded region. [5]

\includegraphics{figure_8} The diagram shows an isosceles triangle \(ACB\) in which \(AB = BC = 8\) cm and \(AC = 12\) cm. The arc \(XC\) is part of a circle with centre \(A\) and radius \(12\) cm, and the arc \(YC\) is part of a circle with centre \(B\) and radius \(8\) cm. The points \(A\), \(B\), \(X\) and \(Y\) lie on a straight line.

1. Show that angle \(CBY = 1.445\) radians, correct to \(4\) significant figures. [3]
2. Find the perimeter of the shaded region. [4]

\includegraphics{figure_1} Figure 1 shows the cross-section ABCD of a chocolate bar, where AB, CD and AD are straight lines and M is the mid-point of AD. The length AD is 28 mm, and BC is an arc of a circle with centre M. Taking A as the origin, B, C and D have coordinates (7, 24), (21, 24) and (28, 0) respectively.

1. Show that the length of BM is 25 mm. [1]
2. Show that, to 3 significant figures, \(\angle BMC = 0.568\) radians. [3]
3. Hence calculate, in mm², the area of the cross-section of the chocolate bar. [5]

Given that this chocolate bar has length 85 mm,

1. calculate, to the nearest cm³, the volume of the bar. [2]

\includegraphics{figure_1} Figure 1 shows a circle of radius \(r\) and centre \(O\) in which \(AD\) is a diameter. The points \(B\) and \(C\) lie on the circle such that \(OB\) and \(OC\) are arcs of circles of radius \(r\) with centres \(A\) and \(D\) respectively. Show that the area of the shaded region \(OBC\) is \(\frac{1}{6}r^2(3\sqrt{3} - \pi)\). [6]