To find the missing side of a right triangle one can use the Pythagorean Theorem Formula: `a^2+b^2=c^2`

When using this formula side a and side b represent the sides of the right triangle and side c represents the hypotenuse of the right triangle. The hypotenuse will always be the longest length of a right triangle. What is not stated in the given information was which side represents the length of hypotenuse. The length 10cm or the length 3x are possible lengths of the hypotenuse. Therefore, the problem will be solved the using both situations.

Situation #1

Let the length 10 cm and length x be the legs of the right triangle and 3x is the length of the hypotenuse. Using the Pythagorean Theorem Formula:

`(10)^2+(x)^2=(3x)^2`

`100+x^2=9x^2`

`100=8x^2`

`100/8=x^2`

` ` `25/2=x^2`

`sqrt(25/2)=x`

`5/sqrt(2)=x`

`(5sqrt(2))/(2)=x`

The value of x is `(5sqrt(2))/(2) cm.`

Situation #2

Let the length x and the length 3x be the legs of the right triangle and 10 cm

is the length of the hypotenuse. Using the Pythagorean Theorem Formula:

`(x)^2+(3x)^2=(10)^2`

`x^2+9x^2=100`

`10x^2=100`

`x^2=10`

`x=sqrt(10)`

The value of x is `sqrt(10) cm.`

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