The Square Under the Hypotenuse
Method 1:
If the square has side length $x$, what are the side lengths of the smaller triangles?
How do we know the triangles are similar?
Using the fact that the triangles are similar, can we write any equations involving $x$?
Method 2:
If we draw the triangle on a coordinate grid, with the right angle at the origin, what are the coordinates of the vertices of the triangle?
How can we work out the equation of the hypotenuse?
What would the equation of the diagonal of the square be?
Method 3:
What is the length and width of the original rectangle?
What are the dimensions of the rearranged rectangle?
Can you use the fact that the areas are the same to write an equation?
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Fitting In
The largest square which fits into a circle is ABCD and EFGH is a square with G and H on the line CD and E and F on the circumference of the circle. Show that AB = 5EF. Similarly the largest equilateral triangle which fits into a circle is LMN and PQR is an equilateral triangle with P and Q on the line LM and R on the circumference of the circle. Show that LM = 3PQ
Triangle Midpoints
You are only given the three midpoints of the sides of a triangle. How can you construct the original triangle?