The Pythagorean Theorem
The Pythagorean theorem states that:
[tex]a^2+b^2=c^2[/tex]
- a and b are two legs of a right triangle
- c is the hypotenuse
The Quadratic Formula
[tex]x=\dfrac{-b\pm \sqrt{b^2-4ac}}{2a}[/tex]
Solving the Question
Let a represent the length of one leg.
Because the hypotenuse is 14 cm longer than a leg, we can say that the hypotenuse's length is 14 + a.
Plug these into the Pythagorean theorem:
[tex]a^2+b^2=c^2\\a^2+a^2=(14+a)^2\\2a^2=14^2+2(14)a+a^2\\2a^2=196+28a+a^2\\a^2=196+28a\\a^2-196-28a=0\\a^2-28a-196=0[/tex]
Factor using the quadratic formula:
[tex]a=\dfrac{-b\pm \sqrt{b^2-4ac}}{2a}[/tex]
[tex]a=\dfrac{-(-28)\pm \sqrt{(-28)^2-4(1)(-196)}}{2(1)}\\\\a=14\pm14\sqrt{2}\\\\a=14+14\sqrt{2}[/tex]
We know that it's plus because subtracting results in a negative value, and length cannot be negative.
This is the length of each side.
Because the hypotenuse is 14 cm longer, we can say that the hypotenuse is [tex]28+14\sqrt{2}[/tex].
Answer
Leg length = [tex]14+14\sqrt{2}[/tex]
Hypotenuse length = [tex]28+14\sqrt{2}[/tex]
