Multiple choice

Passage

Directions: This data sufficiency problem consists of a question and two statements labelled (1) and (2). You have to decide whether the data given in the statements is sufficient for answering the question.

A triangle ABC has a right angle at C. What is the value of (AC + BC)? (1) The radius of the inscribed circle of triangle ABC is equal to 5 cm. (2) The radius of the circumscribed circle of triangle ABC is equal to 9 cm.

  1. Statement (1) alone is sufficient, but statement (2) alone is not sufficient.

  2. Statement (2) alone is sufficient, but statement (1) alone is not sufficient.

  3. Both statements (1) and (2) together are sufficient, but neither of them alone is sufficient.

  4. Each statement alone is sufficient.

  5. Both statements (1) and (2) together are not sufficient.
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

For a right triangle, the inradius r = (a+b-c)/2 and circumradius R = c/2. Statement 1 gives r=5, so a+b-c = 10. Statement 2 gives R=9, so c=18. Together, a+b-18 = 10, so a+b = 28.

AI explanation

Statement (2) gives the hypotenuse AB as twice the circumscribed circle's radius, making AB equal to 18 cm, but this alone cannot determine the sum of the legs AC and BC. Statement (1) provides the inradius as 5 cm, meaning the area is 2.5 times the perimeter, but without the hypotenuse this is insufficient. Using the right triangle inradius formula r = (AC + BC - AB) / 2, substituting r = 5 and AB = 18 yields AC + BC = 28 cm. Therefore, both statements together are sufficient, but neither statement alone is sufficient.

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