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This problem trains for: SAT-I, AMC-8, GMAT, AMC-10.
A triangle has sides AB=10 and BC=24. Which of the following statements is necessary and sufficient in order to decide that the side AC has length 26?
The numbers form a Pythagorean triple:
If the triangle is right, and the 90° angle is opposite from AC then the side AC is the hypotenuse and has length 26 as desired.
The condition " the angle ∠ABC=90° " specifies that the triangle is right and that the right angle is opposite from AC. This makes AC a hypotenuse and the Pythagorean triples apply.
Since the side lengths of 10 and 24 are not in a ratio specific to a special triangle (such as 30°-60°-90°), the fourth condition is not true.
The fifth condition makes AC a leg - this cannot happen since the other two sides are both shorter than it.