Question

In the figure below, find the exact value of z. (Do not approximate your answer.)

Answer 1

`z = (25)/(3)`

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:

`c^2=a^2+b^2`

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:

`h^2=5^2-3^3=25-9=16`

`h=√(16)=4`

Now for the bigger triangle:

`z^2=5^2+x^2`

Solving for
`x^2`

`x^2=z^2-5^2`

For the smaller right-side triangle:

`x^2=4^2+(z-3)^2`

Equating both equations:

`z^2-5^2=4^2+(z-3)^2`

Expanding the square:

`z^2-5^2=4^2+z^2-6z+9`

Simplifying and operating:

`-25=16-6z+9`

`6z=25+25=50`

`z = (50)/(6)=(25)/(3)`

`\boxed{z = (25)/(3)}`