ancient Chinese maths in right angled triangles

a ppt gives the problems and suggests approaches to each problem

there are 24 problems on this 'Gougou' topic
the first 10 are here (the rest are included on the ppt)

I have changed problems 7 and 8 to involve more manageable numbers
(my apologies)

similar triangles

these questions are from or are similar to CBSE (India) Y10 papers

(5) and (6) also need circle theorems

biggest square inside a right angled triangle

maybe an interesting application of similar triangle techniques
but some hefty algebraic manipulation is needed to establish a general result

students could possibly be presented with the derivation of a general result and be asked to sort out what is happening at each stage

unsurprisingly, I subsequently found that this has been explored by (amongst others)
Marion Walter in FLM 21 in 2001 and MT 53 in 1970

her general results agree with those above
(it looks like there is a small numerical error on p.29 of her article)

she points out that 1/y^2 = 1/x^2 + 1/c^2 where 'c' is the hypotenuse

drawing similar triangles

an intention is that students use a 'vectorial' approach to completing the similar triangles

for the first side, the angles of the triangles can be calculated from just this information:

perpendicular to the hypotenuse

thanks to 'five triangles' on twitter  (on jan 7th 2016)

a five-reptile

what are the proportions for the lengths of sides of a right angled triangle if five of them (all congruent) make up a triangle similar to the original?

thanks to John Golden ('math hombre') for pointing out that this forms the basis of the aperiodic 'pinwheel' tiling by John Conway

cone frustum

how do you work out the volume (in general)?
[it's a cone with a smaller cone chopped off]

how do you work out the curved surface area?
[it's a part of an annulus]

what if you don't know 'h' ?

find the volume of the cup

what is the volume of this bowl?

work out the dimensions of the portion of the annulus (washer shape) needed to make the fabric of this lampshade

(ignoring any overlap of material)

show that the angle required is 207.57 (to 2dp) degrees