(a) Evaluate, without using mathematical tables, \(17.57^{2} - 12.43^{2}\). (b) Prove that angles in the same segment of a circle are equal.

Answer Details

(a) Use the difference of two squares, \(a^2 - b^2 = (a-b)(a+b)\):

\(17.57^2 - 12.43^2 = (17.57 - 12.43)(17.57 + 12.43)\)

\(= (5.14)(30.00) = \mathbf{154.2}\)

(b) Theorem: Angles in the same segment of a circle are equal.

Given: A circle with centre \(O\); a chord \(AB\); points \(P\) and \(Q\) on the major arc (the same segment), so \(\angle APB\) and \(\angle AQB\) both stand on chord \(AB\).

To prove: \(\angle APB = \angle AQB\).

Construction: Join \(OA\) and \(OB\).

Proof: By the theorem that the angle subtended by an arc at the centre is twice the angle it subtends at the circumference (on the same arc):

\(\angle AOB = 2\,\angle APB \quad\text{and}\quad \angle AOB = 2\,\angle AQB\)

Therefore \(2\,\angle APB = 2\,\angle AQB\), giving \(\angle APB = \angle AQB\). Hence angles in the same segment are equal. \(\blacksquare\)