Lawnmower

A kite shaped lawn consists of an equilateral triangle ABC of side 130 feet and an isosceles triangle BCD in which BD and CD are of length 169 feet. A gardener has a motor mower which cuts strips of grass exactly one foot wide and wishes to cut the entire lawn in parallel strips. What is the minimum number of strips the gardener must mow?

Bi-cyclics

Two circles intersect at A and B. Points C and D move round one circle. CA and DB cut the other circle at E and F. What do you notice about the line segments CD and EF?

Floating in Space

Stage: 4 Challenge Level:

Many thanks Andrei (School 205, Bucharest) for this well explained solution. A correct solution was also sent by Thomas, whilst Lawrence, of Beecroft Primary, may not have managed the complete journey - he certainly saw the importance of the parallel lines!

Let L be a point on BC and M on YZ so that:
BL$\equiv$ YM, and point K on AB and N on XY so that BK$\equiv$ YN.
As BL is parallel and congruent with YM, BLMY is a parallelogram, and LM is parallel and congruent with BY.

In the same situation, BK is parallel and congruent with YN and BKNY is a parallelogram.

So, KN is parallel and congruent with BY.

From the two relations, I observe that KN is parallel and congruent with LM and KLMN is a parallelogram.

So, KL$\equiv$NM.

I already know that BK$\equiv$ NY and LB$\equiv$ MY, and from these three congruence relations, triangles KBL and NYM are congruent.

So, angles KBL and NYM are equal and so are angles ABC and XYZ.