Question

What is the length of the hypotenuse of a right triangle defined by the points (4, 2), (8, 2), and (8, 4)?

Answer 1

Look at the picture.
Use the Pythagorean theorem:
h - a hypotenuse

`h^2=4^2+2^2\\\\h^2=16+4\\\\h^2=20\to h=√(20)\\\\h=√(4\cdot5)\\\\h=2\sqrt5`
Answer: The length of the hypotenuse is equal 2√5.
Other method.
Use the formula of the length of the segment AB:

`A(x_A;\ y_A);\ B(x_B;\ y_B)\\\\|AB|=√((x_B-x_A)^2+(y_B-y_A)^2)`
We have:

`A(4;\ 2);\ B(8;\ 4)\\\\|AB|=√((8-4)^2+(4-2)^2)=√(4^2+2^2)\\\\=√(16+4)=√(20)=√(4\cdot5)=2\sqrt5`