Angles of a cyclic quadrilateral in two unknowns
Angles of a cyclic quadrilateral in two unknowns
IP 2
10 marks
\(ABCD\) is a quadrilateral inscribed in a circle, with \(\angle DAB=(x+2y)^\circ\), \(\angle ABC=3y^\circ\), \(\angle BCD=5x^\circ\) and \(\angle CDA=(4x+10)^\circ\).
- Find the values of \(x\) and \(y\).[3]
- Explain why \(AC\) is a diameter of the circle.[2]
- Given that \(AC=10\) cm and \(AD=CD\), find the exact length of \(AD\), and find \(\angle BAC\) and \(\angle BCA\). Use your angles to check the value of \(\angle BCD\) found in (a).[3]
- Find the area of \(ABCD\), giving your answer correct to \(3\) significant figures.[2]
Solution:
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Answer:(a) \(x=20\), \(y=30\) (b) \(\angle ABC=90^\circ\) stands on \(AC\) (c) \(AD=5\sqrt{2}\) cm, \(\angle BAC=35^\circ\), \(\angle BCA=55^\circ\); \(55^\circ+45^\circ=100^\circ\) (d) \(48.5\text{ cm}^2\)
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