Answer:
The length of z is [tex]3\sqrt{3}[/tex] unit
Step-by-step explanation:
First Label the diagram as shown in the attachment:
In right [tex]\triangle ABC[/tex];
first find the value of y:
Use Pythagoras theorem states that in a right angled triangle, the square of the hypotenuse is equal to the sum of the squares on the other two sides.
In [tex]\triangle ABC[/tex]
length of AB =6 unit , length of BC =y unit and the length of AC = 3+9 =12 unit
Now, using Pythagoras theorem to get the value of y;
[tex]AB^2+BC^2=AC^2[/tex]
[tex]6^2+y^2=12^2[/tex] or
[tex]36+y^2=144[/tex] or
[tex]y^2=144-36[/tex]=108 or
[tex]y=\sqrt{108}[/tex]
on simplify we get, [tex]y=6\sqrt{3}[/tex] unit
Therefore, the length of BC is [tex]6\sqrt{3}[/tex] unit.
now, in [tex]\triangle BDC[/tex]
using Pythagoras theorem to find the value of z;
[tex]BD^2+CD^2=BC^2[/tex]
Putting the values of BD = z units , BC = [tex]6\sqrt{3}[/tex] unit and DC = 9 units we have,
[tex]z^2+9^2=(6\sqrt{3})^2[/tex] or
[tex]z^2+81=108[/tex] or
Simplify:
[tex]z^2=27[/tex] or
[tex]z=\sqrt{27} = 3 \sqrt{3}[/tex]
Therefore, the value of length z is [tex]3\sqrt{3}[/tex] unit
