In triangle △abc, ∠abc=90°, bh is an altitude. find the missing lengths. ab=4 and bc=3, find ah, ch and bh.

Step-by-step explanation:  

It can be convenient to compute the length of the hypotenuse of this triangle (AC). The Pythagorean theorem tells you ...

   AC^2 = AB^2 + CB^2

   AC^2 = 4^2 + 3^2 = 16 + 9 = 25

   AC = √25 = 5  

The altitude divides ∆ABC into similar triangles ∆AHB and ∆BHC. The scale factor for ∆AHB is ...

  scale factor ∆ABC to ∆AHB = AB/AC = 4/5 = 0.8  

And the scale factor to ∆BHC is ...  

  scale factor ∆ABC to ∆BHC = BC/AC = 3/5 = 0.6  

Then the side AH is 0.8·AB = 0.8·4 = 3.2

And the side CH is 0.6·BC = 0.6·3 = 1.8  

These two side lengths should add to the length AC = 5, and they do.  

The remaining side BH can be found from either scale factor:    

  BH = AB·0.6 = BC·0.8 = 4·0.6 = 3·0.8 = 2.4

_____  

The sides of interest are ...  

  AH = 3.2

  CH = 1.8

  BH = 2.4

the correct answer to your question is a. 4^4

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