## Challenge 236: Magnificent Maximising

**a** A triangle ABC is right angled at B and has base AB of length 24 and height BC of length 6.

A rectangle is drawn with sides parallel to AB and BC, and with one vertex at B, one at P on AB, one at Q on AC and one at R on BC. The height of this rectangle is *a*.

Find the value of *a* which makes the area of the rectangle as large as possible.

**b** A triangle ABC is right angled at B and has base AB of length 24 and height BC of length 6.

A rectangle is drawn with sides parallel to AB and BC, and with one vertex at B, one at P on AB, one at Q on AC and one at R on BC. The height of this rectangle is *a*.

A second rectangle is drawn with sides parallel to AB and BC, and with one vertex at R, one at S on QR, one at T on AC and one at U on BC, one at P on AB, one at Q on AC and one at R on BC. This rectangle has height *b*.

Find the values of *a* and *b* which make the combined area of the two rectangles as large as possible.