three geometrical problems

[kate]

1. The sides of two equilateral triangles are 4cm and 8cm. Find the relationship between the areas of the two triangles.

2. The area of a triangle is 32cm². Find the area of the triangle whose points are middles of the sides of the first triangle.

3. The hypotenuses of two equilateral triangles are 3cm and 12cm. Find the relationship between their areas.

[HallsofIvy]

Area of two "similar" figures (angles are the same, lengths in proportion) are proportional to the square of the distances. In particular, if side lengths are doubled, areas are multiplied by 4. If sides lengths are halved, areas are divided by 4.

(I have no idea what the "hypotenuse" of an equilateral triangle is! I suspect you mean "altitude". If you are, in any case, referring to a length, if one length is 4 times the other the the area is 4^2= 16 times the other.)

[mycalsuite]

Ratio of area= 4^2 : 8^2 = 1:4
New triangle areas = 32/4= 8 cm^2
Ratio of areas = 3^2 : 12^2 = 1:16