Libby wants to measure the length of a pond. She measured 13 yd from point X to point Z and 12 yd from point Y to point Z. What is the length of the pond?

Question

The image depicts a right triangle, labeled XYZ, with a blue, irregular shape obscuring the length of side XY. The angle at vertex Y is marked as a right angle. A dashed line extends from point X perpendicular to side YZ, indicating the height of the triangle relative to base YZ. The question mark on the dashed line suggests that the length of this line is unknown or is what needs to be determined. The triangle's sides are labeled as follows:
XY is the vertical side, partially obscured by the blue shape.
YZ is the horizontal base of the triangle.
XZ is the hypotenuse, connecting points X and Z.
Responses
A 1 yd1 yd
B 5 yd5 yd
C 18 yd18 yd
D 25 yd25 yd
E 30 yd

Answer 1

To find the length of the missing side XY in the right triangle XYZ, we can use the Pythagorean theorem. The theorem states that in a right triangle, the square of the length of the hypotenuse (XZ in this case) is equal to the sum of the squares of the lengths of the other two sides (XY and YZ).

Given:

• XZ (hypotenuse) = 13 yd
• YZ (one leg) = 12 yd
• XY (the other leg) = ?

According to the Pythagorean theorem:

\[ XZ^2 = XY^2 + YZ^2 \]

Substituting the known values:

\[ 13^2 = XY^2 + 12^2 \]

Calculating the squares:

\[ 169 = XY^2 + 144 \]

Now, isolate \( XY^2 \):

\[ XY^2 = 169 - 144 \] \[ XY^2 = 25 \]

Taking the square root of both sides:

\[ XY = 5 , \text{yd} \]

Therefore, the answer is:

B 5 yd