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# Quad in Quad

*Quad in Quad printable sheet*

Printable dotty paper

Draw any convex quadrilateral and find the midpoints of the four edges.

Join the midpoints to form a new quadrilateral.

*A convex quadrilateral is one where all of the angles are less than $180^{\circ}$. Alternatively you can use the definition that both diagonals lie inside the shape.*

Try it a few times starting with different convex quadrilaterals.

What do you notice about the new quadrilaterals formed by joining the midpoints?

Does this always happen?

Can you find a counter-example?

Can you make a conjecture?

You may wish to use the GeoGebra interactivity to explore what happens as you move points A, B, C and D.

Can you prove your conjecture?

Is the area of PQRS always the same fraction of the area of ABCD?

Can you explain why?
*We are very grateful to the Heilbronn Institute for Mathematical Research for their generous support for the development of this resource.*
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Printable dotty paper

Draw any convex quadrilateral and find the midpoints of the four edges.

Join the midpoints to form a new quadrilateral.

Try it a few times starting with different convex quadrilaterals.

What do you notice about the new quadrilaterals formed by joining the midpoints?

Does this always happen?

Can you find a counter-example?

Can you make a conjecture?

You may wish to use the GeoGebra interactivity to explore what happens as you move points A, B, C and D.

Can you prove your conjecture?

Is the area of PQRS always the same fraction of the area of ABCD?

Can you explain why?

Once you have had a think about this, you might like to take a look at these two different proofs which have been scrambled up. Can you rearrange them back into their original order?

Do these results still hold if ABCD is a concave quadrilateral?

*A concave quadrilateral is one where one angle is greater than $180^{\circ}$, for example you could draw an "arrowhead" shape.*

If you can find a proof which is different to the ones in our proof sorters, then please do let us know by submitting it as a solution.

A triangle PQR, right angled at P, slides on a horizontal floor with Q and R in contact with perpendicular walls. What is the locus of P?

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.

Four rods are hinged at their ends to form a quadrilateral. How can you maximise its area?