<p>The diagram below shows a quadrilateral \(ABCD\) with obtuse angles \(A\hat{B}C\) and \(A\hat{D}C\).</p>
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<p>\(AB = 5\) cm, \(BC = 4\) cm, \(CD = 4\) cm, \(AD = 4\) cm, \(B\hat{A}C = 30^{\circ}\), \(A\hat{B}C = x^{\circ}\), \(A\hat{D}C = y^{\circ}\).</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Use the cosine rule to show that&nbsp;\(AC = \sqrt{41 - 40\cos{x}}\).<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Use the sine rule in triangle \(ABC\) to find another expression for&nbsp;\(AC\).<span class="inline-marks">[2]</span></div></li><li><ol class="i-list"><li><div class="list-item-paragraph">Hence, find \(x\), giving your answer to two decimal places.<span class="inline-marks">[6]</span></div></li><li><div class="list-item-paragraph">Find \(AC\).</div></li></ol></li><li><ol class="i-list"><li><div class="list-item-paragraph">Find \(y\).<span class="inline-marks">[5]</span></div></li><li><div class="list-item-paragraph">Hence, or otherwise, find the area of triangle \(ACD\).</div></li></ol></li></ol>

The diagram below shows a quadrilateral \(ABCD\) with obtuse angles \(A\hat{B}C\) and \(A\hat{D}C\).

\(AB = 5\) cm, \(BC = 4\) cm, \(CD = 4\) cm, \(AD = 4\) cm, \(B\hat{A}C = 30^{\circ}\), \(A\hat{B}C = x^{\circ}\), \(A\hat{D}C = y^{\circ}\).

Use the cosine rule to show that \(AC = \sqrt{41 - 40\cos{x}}\).

Use the sine rule in triangle \(ABC\) to find another expression for \(AC\).

Trigonometry

International Baccalaureate Standard Level Mathematics