The cue for the Pythagorean theorem is a right triangle with two side lengths given. If both given sides are legs, add their squares to get the hypotenuse’s square. Then take the positive square root and simplify it. To find a missing hypotenuse, take the square root of the sum of the legs’ squares. Adding the leg lengths themselves won’t work.
Hints
- Hint 1
The hypotenuse is the side across from the right angle; the two legs meet at that angle. In the Pythagorean theorem, which side stands alone as ?
- Hint 2
For two known legs, the Pythagorean theorem says to add their squares. That gives you the hypotenuse squared. What must you do next to get its length?
- Hint 3
An exact radical keeps the square root instead of rounding it. Look for a perfect-square factor inside the root so you can simplify the length.
Step-by-step
Add the squares, then simplify
Step 1Write the equation for the hypotenuse
The - and -centimeter sides are legs, the sides that meet at the right angle. Let be the hypotenuse, the side across from that angle. The Pythagorean theorem says the legs’ squares add to the hypotenuse’s square, so:
- Step 2
Find the square of the length
Type into Desmos. It displays , so . Don’t stop at : it’s the square of the hypotenuse’s length, not the length.
- Step 3
Take the positive square root
Because is a length, take the positive square root to undo the square:
- Step 4
Expose a perfect-square factor
Use to expose the perfect-square factor :
- Step 5
Simplify the exact length
Take outside the radical:
The hypotenuse is centimeters. Choice B.