Answer:
[tex]z = \frac{25}{3}[/tex]
Step-by-step explanation:
Pythagora's Theorem
In any right triangle, the square of the measure of the hypotenuse is the sum of the squares of the legs. This can be expressed with the formula:
[tex]c^2=a^2+b^2[/tex]
Where
c = Hypotenuse or largest side
a,b = Legs or shorter sides
We are required to find z in the figure provided. We have completed the construction with two additional variables h and x in the image below.
The triangle to the left side has hypotenuse 5 and one leg of 3, thus:
[tex]h^2=5^2-3^3=25-9=16[/tex]
[tex]h=\sqrt{16}=4[/tex]
Now for the bigger triangle:
[tex]z^2=5^2+x^2[/tex]
Solving for [tex]x^2[/tex]
[tex]x^2=z^2-5^2[/tex]
For the smaller right-side triangle:
[tex]x^2=4^2+(z-3)^2[/tex]
Equating both equations:
[tex]z^2-5^2=4^2+(z-3)^2[/tex]
Expanding the square:
[tex]z^2-5^2=4^2+z^2-6z+9[/tex]
Simplifying and operating:
[tex]-25=16-6z+9[/tex]
[tex]6z=25+25=50[/tex]
[tex]z = \frac{50}{6}=\frac{25}{3}[/tex]
[tex]\boxed{z = \frac{25}{3}}[/tex]
