A right-angled triangle with a line perpendicular to the hypotenuse

                                                                         

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Question

Here is a right-angled triangle has an extra line descending from the original right angle, perpendicular to the hypotenuse.

If the hypotenuse (base) length is 7cm and if that extra line splits the base into 3cm and 4cm parts, what is the height of 'big' right-angled triangle?

Hint: (From nRICH)
Sometimes we use the word "scale" to mean enlarge or scale up - for another way to see this arrangement of right-angled triangles you may like to look at the problem called a Matter of Scale.




Answer

Scale Factor (*a)
BD = ab
AB = ac
AD = a^2 = 3


a = square root (3)


Scale Factor (*b)
DC = b^2 = 4 
BD = ab
BC = bc


b = square root (4)
b = 2


So: a * b
square root (3) * 2


= 2square root (3)
= 3.4641
BD = 3.46 (3.s.f)