Triangles in a Square
Kira Dhariwal Year 7 Wycombe High School
The blue triangle looks like it has the largest area.
How to calculate the area of each triangle.
When the triangles are drawn on the dotty paper they are surrounded by 3 right-angled triangles. The total area of all three right-angled triangles must be subtracted from the area of the square (25 squares) to give the area of the triangle.
Red triangle is 10.5 squares
Yellow triangle is 10.5 squares
Blue triangle is 11.5 squares
Green triangle is 11 squares
Purple triangle is 10.5 squares
Therefore the blue triangle has the largest area
A triangle is formed with a vertex at (5,5), a vertex on the left hand side and a vertex on the bottom side of the square. What is the smallest area such a triangle can have?
The smallest area the triangle can have is 2.5 squares.
This is when the vertices are (5,5) (0,1) and (0,0) or (5,5) (0,0) and (1,0)
What about the largest area?
The largest area the triangle can have is 12.5 squares. There are multiple ways it can be drawn.
Three examples are vertices
(5,5) (0,5) and (5,0)
or (5,5) (0,0) and (5,0)
or (5,5) (0,5) and (0,0)
Which areas in between is it possible to make?
The areas are 4.5, 5, 6.5, 7.5, 8, 8.5, 9.5, 10, 10.5, 11, 11.5, 12 squares. I drew all the different triangles and calculated the areas to find these answers
How many of these areas are whole numbers?
Five of the areas are whole numbers. This is the same as the number of squares up or down on the 5 by 5 grid.
I predict that on a 6 by 6 grid 6 of the possible areas are whole numbers and on a 7 by 7 grid 7 of the areas are whole numbers etc…..
Can you find a general expression for the area of a triangle on this grid if its vertices have co-ordinates (5,5), (x,0) and (0,y)?
Area square is 5x5= 25 squares
Triangle A has vertices (5,5), (0,5) and (0,y)
Area triangle A = 1/2 x 5 x (5-y) = 2.5(5-y) =12.5 -2.5y
Triangle B has vertices (5,5), (5,0) and (x,0)
Area triangle B = 1/2 x 5 x (5-x) = 2.5(5-x) = 12.5 - 2.5x
Triangle C has vertices (0,0), (0,y) and (x,0)
Area triangle C = 1/2xy =0.5xy
Total area of triangles A+ B+C= 12.5 -2.5x + 12.5 -2.5y + 0.5xy = 25 - 2.5x -2.5y + 0.5xy
Therefore area of triangle drawn in square = 25 – area (trA + trB + trC)
= 25 - ( 25 - 2.5x -2.5y + 0.5xy)
=2.5x+ 2.5y - 0.5 xy squares
What can you say about the areas of triangles drawn on a 6 by 6 grid? Or a 7 by 7 grid? Or a 100 by 100 grid...?
When a triangle is drawn in a 6 by 6 grid the area can be represented by
3x + 3y – 0.5xy
When a triangle is drawn in a 7 by 7 grid the area can be represented by
3.5x + 3.5y – 0.5xy
This shows that there is a pattern for finding the area of the triangle. X and y are multiplied by half of the number of squares up or down.
Let n represent size of grid eg 7 by 7 grid is when n is 7 or 6 by 6 grid is when n is 6
Area of triangle in square = 1/2nx + 1/2 ny - 1/2xy squares
Therefore when a triangle is drawn in a 100 by 100 grid the area can be represented by 50x + 50y – 0.5xy