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This right-angled triangle has a base of 3 and a height of 6 units.
How might you construct the square, which just touches the hypotenuse?
Can you work out the side length of the square?
Can you think of more than one way to work it out?
What if the side lengths of the triangle were 12 and 4 units long?
What if they were $a$ and $b$ units long?
Once you've had a go at solving this, click below to reveal three different approaches.
Can you take each starting point and turn it into a solution?
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
Can you spot a cunning way to work out the missing length?
You are only given the three midpoints of the sides of a triangle. How can you construct the original triangle?