Cool Geometry Trick

Geometry Level pending

A right triangle with all even integer side lengths is inscribed in a circle with area $b\pi$

Which of the following is true about the value of $\sqrt{b}$ ?

Details and Assumptions

$\sqrt{b}$ = $a$ .

No lengths of the triangle can be 0 (or it wouldn't be a triangle).

a

is a square number None of the other options are consistently true

a

is a number from 1 to 100

a

is strictly between 1 and 1 googol (

10^{100}

)

a

is not natural

a

is not strictly between 1 and 1 googol (

10^{100}

)

a

is a natural number

a

is not a square number

1 solution

Clive Chen
Jun 14, 2015

If a right triangle is inscribed in a circle, its hypotenuse is the circle's diameter. Since the length of the hypotenuse is even, the length of the circle's diameter is even, implying that the length of the circle's radius is natural.

Area of a circle = $\pi r^{2}$

Since $r$ is natural, $r^{2}$ is natural, so $a$ is natural.

Isn't $a$ also a square number?

Alex Li - 5 years, 12 months ago

Cool Geometry Trick

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