Right triangle transformation and new hypotenuse: A right-angled triangle has hypotenuse 10 cm and area 24 cm^2. If the shorter leg is halved and the longer leg is doubled, find the length of the new hypotenuse.

Introduction / Context:This problem uses two invariants of a right triangle—Pythagoras and area—to recover the legs, then applies specified changes to the legs and recomputes the hypotenuse.

Given Data / Assumptions:

• Right triangle with legs a and b, hypotenuse 10 cm.
• Area = 24 cm^2 → (1/2) * a * b = 24 → ab = 48.
• Pythagoras: a^2 + b^2 = 10^2 = 100.
• Shorter leg is halved; longer leg is doubled.

Concept / Approach:From ab and a^2 + b^2, obtain (a + b)^2 = a^2 + b^2 + 2ab = 100 + 96 = 196 → a + b = 14. Solve a and b from t^2 − 14t + 48 = 0.

Step-by-Step Solution:

Verification / Alternative check:Original: ab = 48 and a^2 + b^2 = 100 hold for 6 and 8. Transform as per instruction; recomputation matches √265 exactly.

Why Other Options Are Wrong:√245, √255, √275, √225 are not equal to √(3^2 + 16^2).

Common Pitfalls:Halving/doubling the wrong legs or attempting to scale the hypotenuse directly rather than recomputing with Pythagoras after the change.

Final Answer:√265cm