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# Isometric Areas

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### Isosceles

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Links to the NRICH website Home page

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Age 11 to 14

Challenge Level

Here is an equilateral triangle with sides of length 1.

Let's define a unit of area, $T$, such that the triangle has area $1T$.

Here are some parallelograms whose side lengths are whole numbers.

Can you find the area, in terms of $T$, of each parallelogram?

Compare the results with the lengths of their edges.

**What do you notice?
Can you explain what you've noticed?**

Can you find a similar result for trapeziums in which all four lengths are whole numbers?

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?