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Kissing Triangles

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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.


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Isosceles Triangles

Age 11 to 14
Challenge Level

Yanqing from Devonport High School for Girls sent us the only correct solution to this problem. She wrote:

First, I worked out how many types of isosceles triangles there are that have an area of 9cm ², and can be drawn with integer co-ordinates.

I found 3 types:

base = 18, height = 1,
base = 6, height = 3,
base = 2, height = 9.

(These would not work the other way round because it would be impossible to have integer co-ordinates.)

I then worked out that because there are 3 vertices on a triangle, and there are 4 directions in which a triangle could go, it would be possible to have 12 ways of placing one type of triangle around the point (20,20): you can place it 3 ways upwards, 3 ways to the left, to the right and down.

I then tried this out and found it to be the case.

Because there are 3 types of isosceles triangles, and 12 ways to place each one, it is possible to draw 36 isosceles triangles with integer co-ordinates and one vertex on (20,20) which have an area of 9cm ².

Well done Yanqing - this is a very clear explanation.