Given area find angle/side

1. The triangle \(A B C\) is such that

• \(A B = 15 \mathrm {~cm}\)
• \(A C = 25 \mathrm {~cm}\)
• angle \(B A C = \theta ^ { \circ }\)
• area triangle \(A B C = 100 \mathrm {~cm} ^ { 2 }\)
  1. Find the value of \(\sin \theta ^ { \circ }\)

Given that \(\theta > 90\)

find the length of \(B C\), in cm , to 3 significant figures.

1 \includegraphics[max width=\textwidth, alt={}, center]{555f7205-5e2a-4471-901d-d8abc9dd4b4a-2_250_611_356_721} The diagram shows triangle \(A B C\), with \(A C = 8 \mathrm {~cm}\) and angle \(C A B = 30 ^ { \circ }\).

1. Given that the area of the triangle is \(20 \mathrm {~cm} ^ { 2 }\), find the length of \(A B\).
2. Find the length of \(B C\), giving your answer correct to 3 significant figures.

6. Figure 1 Figure 1 shows a sketch of a triangle \(A B C\) with \(A B = 3 x \mathrm {~cm} , A C = 2 x \mathrm {~cm}\) and angle \(C A B = 60 ^ { \circ }\) Given that the area of triangle \(A B C\) is \(18 \sqrt { 3 } \mathrm {~cm} ^ { 2 }\)

1. show that \(x = 2 \sqrt { 3 }\)
2. Hence find the exact length of BC, giving your answer as a simplified surd.

4 The triangle \(A B C\), shown in the diagram, is such that \(A B\) is 10 metres and angle \(B A C\) is \(150 ^ { \circ }\). \includegraphics[max width=\textwidth, alt={}, center]{02e5dfac-18d7-480d-ac23-dfd2ca348cba-3_323_746_406_648} The area of triangle \(A B C\) is \(40 \mathrm {~m} ^ { 2 }\).

1. Show that the length of \(A C\) is 16 metres.
2. Calculate the length of \(B C\), giving your answer, in metres, to two decimal places.
3. Calculate the smallest angle of triangle \(A B C\), giving your answer to the nearest \(0.1 ^ { \circ }\).

3 The diagram shows a triangle \(A B C\). \includegraphics[max width=\textwidth, alt={}, center]{bfe96138-9587-4efb-95c5-84c4d5eadfbe-3_273_622_356_708} The lengths of \(A C\) and \(B C\) are 5 cm and 6 cm respectively.
The area of triangle \(A B C\) is \(12.5 \mathrm {~cm} ^ { 2 }\), and angle \(A C B\) is obtuse.

1. Find the size of angle \(A C B\), giving your answer to the nearest \(0.1 ^ { \circ }\).
2. Find the length of \(A B\), giving your answer to two significant figures.

2 The diagram shows a triangle \(A B C\) in which angle \(C = 30 ^ { \circ } , B C = x \mathrm {~cm}\) and \(A C = ( x + 2 ) \mathrm { cm }\). Given that the area of triangle \(A B C\) is \(12 \mathrm {~cm} ^ { 2 }\), calculate the value of \(x\).

2 \includegraphics[max width=\textwidth, alt={}, center]{ac5bf967-8b97-4bf3-991f-28c3ec7a25da-2_399_933_968_561} The diagram shows a triangle \(A B C\) in which angle \(C = 30 ^ { \circ } , B C = x \mathrm {~cm}\) and \(A C = ( x + 2 ) \mathrm { cm }\). Given that the area of triangle \(A B C\) is \(12 \mathrm {~cm} ^ { 2 }\), calculate the value of \(x\).

2 \includegraphics[max width=\textwidth, alt={}, center]{1c957cfe-bead-41d9-8985-479e876e1616-2_403_938_964_559} The diagram shows a triangle \(A B C\) in which angle \(C = 30 ^ { \circ } , B C = x \mathrm {~cm}\) and \(A C = ( x + 2 ) \mathrm { cm }\). Given that the area of triangle \(A B C\) is \(12 \mathrm {~cm} ^ { 2 }\), calculate the value of \(x\).

2 \includegraphics[max width=\textwidth, alt={}, center]{48b63de9-f022-4881-a187-f08e3c7d9f1a-2_399_940_952_561} The diagram shows a triangle \(A B C\) in which angle \(C = 30 ^ { \circ } , B C = x \mathrm {~cm}\) and \(A C = ( x + 2 ) \mathrm { cm }\). Given that the area of triangle \(A B C\) is \(12 \mathrm {~cm} ^ { 2 }\), calculate the value of \(x\).

2 \includegraphics[max width=\textwidth, alt={}, center]{8a0a6e46-99cf-4217-93ad-5ed6e9d7c4ef-2_401_949_959_557} The diagram shows a triangle \(A B C\) in which angle \(C = 30 ^ { \circ } , B C = x \mathrm {~cm}\) and \(A C = ( x + 2 ) \mathrm { cm }\). Given that the area of triangle \(A B C\) is \(12 \mathrm {~cm} ^ { 2 }\), calculate the value of \(x\).

In a triangle \(PQR\), \(PQ = 20\) cm, \(PR = 10\) cm and angle \(QPR = \theta\), where \(\theta\) is measured in degrees. The area of triangle \(PQR\) is 80 cm\(^2\).

1. Show that the two possible values of \(\cos \theta = \pm \frac{3}{5}\) [4]

Given that \(QR\) is the longest side of the triangle,

1. find the exact perimeter of the triangle \(PQR\), giving your answer as a simplified surd. [3]

The diagram below shows a triangle \(ABC\) with \(AC = 5\) cm, \(AB = x\) cm, \(BC = y\) cm and angle \(BAC = 120°\). The area of the triangle \(ABC\) is \(14\) cm\(^2\). \includegraphics{figure_14} Find the value of \(x\) and the value of \(y\). Give your answers correct to 2 decimal places. [6]