To solve for the length of the hypotenuse [tex]\( x \)[/tex] in a right triangle, you can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse ([tex]\( x \)[/tex]) is equal to the sum of the squares of the other two sides. In this case, the two given sides are 16 and 63.
We need to find the equation that correctly uses the Pythagorean theorem. Here are the given choices:
1. [tex]\( 16 + 63 = x \)[/tex]
2. [tex]\( 18^2 + 83 - x \)[/tex]
3. [tex]\( (16 + 63)^2 = x^2 \)[/tex]
4. [tex]\( 16^2 + 63^2 = x^2 \)[/tex]
Now, let’s examine each choice:
1. [tex]\( 16 + 63 = x \)[/tex]
This choice simply adds the lengths of the two sides and states that [tex]\( x \)[/tex] is equal to this sum. This is incorrect because it does not follow the Pythagorean theorem, which involves squaring the sides, not adding them directly.
2. [tex]\( 18^2 + 83 - x \)[/tex]
This choice does not correctly represent the sides of the triangle we are working with. It includes incorrect values (18 and 83) and an incorrect form for the equation involving [tex]\( x \)[/tex]. So, this is incorrect.
3. [tex]\( (16 + 63)^2 = x^2 \)[/tex]
This choice adds the lengths of the two sides and then squares the result to find [tex]\( x^2 \)[/tex]. This is incorrect because it does not correctly apply the Pythagorean theorem, which requires squaring the lengths of the sides individually before summing them.
4. [tex]\( 16^2 + 63^2 = x^2 \)[/tex]
This choice correctly follows the Pythagorean theorem: the square of the hypotenuse ([tex]\( x \)[/tex]) is equal to the sum of the squares of the other two sides (16 and 63).
Therefore, the correct equation to find [tex]\( x \)[/tex], the length of the hypotenuse of the right triangle, is:
[tex]\[ 16^2 + 63^2 = x^2 \][/tex]
This is the appropriate use of the Pythagorean theorem.
