Final answer:
To find the length of segment XW in the right-angled triangle XYZ with parameters given, we use the Pythagorean theorem twice. The correct length of XW is approximately 7.75, which does not match any of the provided options.
Step-by-step explanation:
The student is asked to find the length of segment XW in triangle XYZ, where △XYZ is right-angled at X and line segment W is drawn on YZ such that ∠XWY is 90 degrees. To solve this, we will use the Pythagorean theorem which states that in a right-angled triangle, the square of the length of the hypotenuse (side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
In this case, XY = 8 and XZ = 6, so:
- XY2 + XZ2 = YZ2
- 82 + 62 = YZ2
- 64 + 36 = YZ2
- 100 = YZ2
- 10 = YZ (since we are looking for the positive length of side YZ)
Furthermore, since ∠XWY is a right angle, △XWY is also a right-angled triangle, and WY can be found using the Pythagorean theorem with XY and XW being the legs of the right triangle.
We know that:
- XY2 = XW2 + WY2
- 82 = XW2 + WY2
- 64 = XW2 + WY2
Since YZ = 10 and ∠XWY is a right angle, WY is simply YZ - XY = 10 - 8 = 2. We can now solve for XW:
- 64 = XW2 + 22
- 64 = XW2 + 4
- 60 = XW2
- √60 = XW
- XW = √60 ≈ 7.75 (which is not one of the provided options)
Therefore, the correct length of XW does not match any of the provided options.