Four rods are hinged at their ends to form a convex quadrilateral.
Investigate the different shapes that the quadrilateral can take.
Be patient this problem may be slow to load.
Given a square ABCD of sides 10 cm, and using the corners as
centres, construct four quadrants with radius 10 cm each inside the
square. The four arcs intersect at P, Q, R and S. Find the area
enclosed by PQRS.
A rhombus PQRS has an angle of 72 degrees. OQ = OR = OS = 1 unit. Find all the angles, show that POR is a straight line and that the side of the rhombus is equal to the Golden Ratio.
This problem requires the solver to reason geometrically and make use of symmetry. By re-presenting the information in a different way, for example by adding additional lines (a useful technique in geometrical problems) more structure can be revealed. It is an interesting idea that adding something, and therefore
apparently making it more complex, can sometimes make a problem more accessible. Then of course there is an opportunity to use the cosine rule in a non-standard context.
Use the images in this document to make cards.
Display the problem and ask learners to work in groups rearranging the cards in ways which help to make connections for them.
Share ideas and relationships that groups notice before going on to solve the problem. Encourage careful reasoning and convincing arguments. For example:
There are several ways of solving the problem so share different approaches and discuss what helped to move thinking forward.
Two observations may be worth drawing attention to :
Ask the students if they can explain why these should all be equal .