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Given right triangle ABCwith altitude BD drawn to hypotenuse AC. If AD=4 and DC=8, what is the length of BD? (Note: the figure is not drawn to scale.)

 Sep 24, 2021
 #1
avatar+12422 
+1

What is the length of BD?

 

Hello Guest!

 

Euclid's theorem of height says

\(h^2 = q \times p\)

so

\(\overline{BD}^2=\overline{AD}\times \overline{CD}\\ \overline{BD}^2=4\cdot 8\\ \overline{BD}=\sqrt{32}=\sqrt{2\cdot 16}\)

\(\overline{BD}=4\sqrt{2}=5.657\)

 

The length of BD is 5.657

laugh  !

 Sep 25, 2021
edited by asinus  Sep 25, 2021
edited by asinus  Sep 25, 2021
edited by asinus  Sep 25, 2021
 #3
avatar+1565 
+2

BD = sqrt(DC2 / 2)

civonamzuk  Sep 25, 2021
 #4
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This is true only if  AD : DC = 1 : 2

 

Let's call this Civonamzuk's theorem. laugh

Guest Sep 26, 2021
 #2
avatar+1565 
+1

Triangles ABD and BCD are similar!!!

 

4 / BD = BD / 8

 

BD = √32  = 4√2 smiley

 Sep 25, 2021

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