Quadrilaterals in a square
What's special about the area of quadrilaterals drawn in a square?
Problem
Quadrilaterals in a Square printable sheet
Quadrilaterals in a Square downloadable slides
Suppose we have a yellow square of side length $a+b$.
We can draw quadrilaterals in this square so that one vertex lies on each side of the square, and cuts each side into one segment of length $a$ and one segment of length $b$, as below:
Can you prove that in each of these images the area of the red quadrilateral is exactly half the area of the yellow square?
Try to find two different ways to prove it - one algebraic, and one geometric.
Here are two more images showing quadrilaterals drawn on the yellow square.
Can you prove that the areas of these two red quadrilaterals sum to the area of the yellow square?
Again, try to prove this in both an algebraic and a geometric way.
With thanks to Don Steward, whose ideas formed the basis of this problem.
Getting Started
Algebraic:
Can you find the area of the yellow square?
Can you find the area of the yellow triangles?
Can you use this to find the red area?
Geometric:
Are there any extra lines you could draw?
Student Solutions
Sam from Huntingdon School, Chenthuran from Chamblee Charter High School, Nasif from Wilson's School, Zach, Pablo from Kings College Alicante, Katie at Dulwich College Seoul, Aryan from Dumpton School and Matthew from Lochcarron Primary School all sent in some great work, and thought really hard about the problem. Thank you everybody!
Pablo from Kings College Alicante submitted a solution here, proving algebraically and geometrically the first and third parts of the question.
Thank you Pablo!
Here is another solution, from Katie at Dulwich College Seoul, proving all three parts of the question.
Thank you Katie!
Here is a solution from Rubaiyat from Wilson's School, which addresses each part of the question in a very clear way.
Thank you Rubaiyat!
Teachers' Resources
Why do this problem?
This problem gives students the opportunity to prove a relationship in two different ways; algebraically, and geometrically. It follows on nicely from Triangles in a Square.
Possible approach
This printable worksheet may be useful: Quadrilaterals in a Square.
The diagrams from the problem are available as slides.
Ask students to choose two different numbers between 1 and 8, and to construct the diagram below on squared paper using their two numbers a and b:
"Can you work out the area of the quadrilateral formed in the square?"
Once students have had a chance to work out the area, collect on the board in two columns the area of the quadrilateral and the area of the square.
"What do you notice?" The quadrilateral has half the area of the square.
"Can we prove that it's true for every quadrilateral formed in this way?"
Students could work in pairs to try to come up with a proof. Encourage them to look for an algebraic and a geometric proof. The Solution has some examples of proofs.
Bring the class together and invite a couple of students to present their proofs. Then provide the second diagram:
"Can you prove a similar result for this quadrilateral?"
Finally, students could work on the last part of the problem, showing that the two red areas sum to the area of the yellow square in the diagram below:
Key questions
Can you express each area algebraically?
Are there any lines you could draw?
Are there any shapes you could move in order to compare areas?
Possible extension
Trapezium Four invites students to consider areas of triangles in a trapezium.
Possible support
Triangles in a Square is a simpler problem where students might use similar strategies to find areas.