Question 11
The side lengths of a right-angled triangle are in geometric progression, and the smallest side has length $$2$$ units. The length of the hypotenuse is
Solution
Write the sides as $$2,2r,2r^2$$ with $$r>1$$. The Pythagorean theorem gives $$4+4r^2=4r^4$$, so $$r^4-r^2-1=0$$ and $$r^2=\frac{1+\sqrt5}{2}$$. The hypotenuse is therefore $$2r^2=1+\sqrt5$$.
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