Question

Triangle $ABC$ is a right-angled triangle. $D$ and $E$ are the midpoints of $AB$ and $BC$ respectively. Read the following statements.
I. $AE = 19$ II. $CD = 22$ III. $\angle B$ is a right angle.
Which of the following statements would be sufficient to determine the length of $AC$?

• A. Statement I and Statement II.
• B. Statement I and Statement III.
• C. Statement II and Statement III.
• D. Statement III alone.
• E. All three statements.

Hint:

For problems involving medians and side lengths, combine the {Apollonius Theorem} with any special-angle information (like a right angl(E) to close the system and determine unknown sides.

Answer 1

Answer: (E)

Step 1: Set notation and use the right angle.
Let a = BC, b = AC, c = AB (so b is the hypotenuse if ∠B = 90° from Statement III).
Hence, by Pythagoras, a² + c² = b². (1)

Step 2: Express medians in terms of sides (Apollonius).
Median from A to BC is AE and from C to AB is CD. For any triangle,
AE² = (2b² + 2c² − a²)/4,    CD² = (2a² + 2b² − c²)/4.
Using Statements I and II: AE = 19 ⇒ AE² = 361, CD = 22 ⇒ CD² = 484. Thus,
361 = (2b² + 2c² − a²)/4,    484 = (2a² + 2b² − c²)/4.   (2)

Step 3: Solve for the sides using (1) and (2).
Solve the three equations for a², b², c²:
(a², b², c²) = (420, 676, 256) ⇒ b = √676 = 26.
Hence AC = b = 26 is uniquely determined when all three statements are used.

Step 4: Check insufficiency of any subset.
I & III only: From 361 = (2b² + 2c² − a²)/4 and a² + c² = b², we get 361 = a²/4 + c²; infinitely many (a, c) fit ⇒ b not fixed.
II & III only: From 484 = (2a² + 2b² − c²)/4 and a² + c² = b², we get 484 = a² + c²/4; again infinitely many solutions.
I & II only: Two equations in three unknowns (a, b, c) without the right-angle relation ⇒ b not determined.
Therefore, only Statements I, II, and III together suffice.

Final Answer: Option (E) is sufficient; in fact AC = 26.