Right Triangles and Quadratic Equations
A Right Triangle has one 90 degree angle in it.
The Pythagorean theorem applies ONLY to right triangles.
It states:
(LEG)2 + (LEG)2 = (Hypotenuse)2
Hypotenuse
Leg
Leg
Often the hypotenuse is labelled “c”, and the legs are labelled “a” or “b”, so the equation has come to
be known as: 𝑎2 + 𝑏 2 = 𝑐 2
There are two special right triangles: 30-60-90 and 45-45-90
In a 45-45-90 Right Triangle, the LEGs both measure the SAME.
For example,
4
4
Thus we can write a quadratic equation: 42 + 42 = 𝑐 2
Which becomes: 16 + 16 = 32 = 𝑐 2
Solving this equation, produces the length of the hypotenuse: 𝑐 = ±√32 = ±√16√2 = ±4√2
Only because a hypotenuse is never negative as a length, we state just the positive answer: 4√2
In a 30-60-90 Right Triangle, the side opposite the 30 degree angle is always one-half of the length of
the hypotenuse.
For example,
4
2: side opposite the 30 degree angle.
o
30
Thus we can write a quadratic equation: 𝑎2 + 22 = 42
Simplifying and Solving this equation, produces: 𝑎2 + 4 = 16
𝑎2 + 4 − 4 = 16 − 4
𝑎2 = 12
𝑎 = ±√12 = ±√4√3 = ±2√3
Only because a side is never negative as a length, we state just the positive answer: 2√3