Illustrate by a figure the truth of the identity \[ a^2-b^2 = (a-b)(a+b). \] \item[*3.] If a straight line touch a circle and from the point of contact a chord be drawn, prove that the angles which this chord makes with the tangent are equal to the angles in the alternate segments.
\(AOB, COD\) are two chords of a circle; shew that the triangles \(AOD, COB\) are similar and hence that \(AO \cdot OB = CO \cdot OD\).
An article when sold at a profit of 13 per cent. yields 1s. 5d. more profit than when sold at a profit of \(4\frac{1}{2}\) per cent.; find the prime cost of the article.
By investing in \(5\frac{1}{2}\) per cent. shares a profit of 5 per cent. is obtained on the money invested what is the price of the shares?
A train travels 525 miles; if its average rate had been \(2\frac{1}{2}\) miles per hour faster, it would have taken 1 hour less; find its average rate.
Solve the equations
In an Arithmetic Progression the 9th term is 7 times the 1st term and the sum of the 4th and 6th terms is 32; find the series.