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# Triangles in a Square

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Age 11 to 14

Challenge Level

*Triangles in a Square printable sheet*

*You may wish to print a dotty grid to work on this problem.*

Here is a triangle drawn on a $5$ by $5$ dotty grid by joining the top-right-hand dot to a dot on the left hand side of the grid, and a dot on the bottom of the grid.

Here are some more triangles drawn in the same way.

**Which has the largest area?**

Now, think about the different triangles that can be formed with a vertex at $(5,5)$, a vertex on the left hand side and a vertex on the bottom of the grid.

What is the smallest area such a triangle can have?

What about the largest area?

Which areas in between is it possible to make?

**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 $(0,y)$?**

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...?

*With thanks to Don Steward, whose ideas formed the basis of this problem.*

Prove that a triangle with sides of length 5, 5 and 6 has the same area as a triangle with sides of length 5, 5 and 8. Find other pairs of non-congruent isosceles triangles which have equal areas.

Four rods, two of length a and two of length b, are linked to form a kite. The linkage is moveable so that the angles change. What is the maximum area of the kite?