Given two right triangles where one leg measures $5$, the hypotenuse measures $13$, and the other leg measures $x$, for the triangles to be congruent by the Hypotenuse-Leg (HL) theorem, what must be the value of $x$?

• A. $12$
• B. $8$
• C. $9$
• D. $10$

Answer: (A)

Verified step by step guidance

1. Recall that the Hypotenuse-Leg (HL) theorem states that two right triangles are congruent if their hypotenuse and one corresponding leg are congruent.
2. Identify the given parts: the hypotenuse is 13, one leg is 5, and the other leg is unknown, labeled as \( x \).
3. Use the Pythagorean theorem for right triangles, which states that \( a^2 + b^2 = c^2 \), where \( c \) is the hypotenuse and \( a \), \( b \) are the legs.
4. Substitute the known values into the Pythagorean theorem: \( 5^2 + x^2 = 13^2 \).
5. Solve the equation for \( x \) by isolating \( x^2 \) and then taking the square root to find the length of the other leg.