Final answer:
Using the Pythagorean theorem, we can find the unknown sides of a right triangle by relating the legs and the hypotenuse. Each part of the question is solved by setting up the equation a² + b² = c² with the given lengths and solving for the missing side.
Step-by-step explanation:
Finding Triangle Lengths Using the Pythagorean Theorem
The Pythagorean theorem is essential in solving problems relating to the lengths of sides in a right triangle. Let's solve each part of the question using this theorem:
- For part a, with XW=3 and WY=12, to find WZ, we use the theorem a² + b² = c². Since XW and WY are the legs of the triangle and WZ is the altitude, we have 3² + WZ² = 12², thus, WZ = √(12² - 3²).
- For part b, given WZ=4 and XW=2, to determine WY, we set up the equation 2² + 4² = WY² and solve for WY.
- For part c, XY is the hypotenuse at 22, and XW is a leg at 9. Using the Pythagorean theorem, we find WZ by solving √(22² - 9²).
- For part d, to find XY, with given XZ=10 (leg) and XW=4 (altitude), we apply the theorem, giving us XY = √(10² + 4²).
- For part e, to find XZ when WZ=18 and XW=9, we again use the theorem: 9² + 18² = XZ² and solve for XZ.
- Finally, for part f, with WZ=4 and XY=10 (hypotenuse), and considering the base XW=x, we apply the Pythagorean theorem to find ZY = √(10² - x²).
These calculations are based on the fundamental principle that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.