Lesson: Data Sufficiency Challenging - 22t09

Unexpected Combinations: Example 1

What is the probability that the length of AC is less than or equal to ?

1) The area of circle O is 81.

2) The length of AB is 6.

This question deals with two different types of content - probability and geometry. To answer it, we must figure out whether we can determine a probability relationship, so let's start there.

The definition of probability appears below. Use it to determine what we must focus on to solve this question.

ow we must focus on the possible lengths of AC, and we must examine the diagram to do so. Use the diagram to answer the following questions. When you’re ready, click Continue.

• How is AC related to the rest of the diagram?
• What is the relationship between AC and the length ?

Review Statement 1 below. Is it sufficient or insufficient? Select the correct answer and then click Continue.

1) The area of circle O is 81.

Sufficient

Insufficient

Review Statement 2 below. Is it sufficient or insufficient? Select the correct answer and then click Continue.

2) The length of AB is 6.

Sufficient

Insufficient

Review the statements in combination. Are they sufficient or insufficient? Select the correct answer and then click Continue.

1) The area of circle O is 81.

2) The length of AB is 6.

Sufficient

Insufficient

We know from Statement 2 that the length of AB is 6, so we know that AB² = 36. If AB² = 36, then in order for AC to be less than or equal to , the maximum length of BC would have to be 4, since 4² is 16 and 36 + 16 = 52. If BC were greater than 4, AC would have to be greater than . So the probability that AC has a length less than or equal to must be same as the probability that BC has a length that’s less than or equal to 4.

If B is a point on the diameter of circle O somewhere between point C and point O (the center of the circle), then BC has a maximum length of 9 (the length of the radius of the circle). But only of that length would produce a hypotenuse AC with a length that’s less than or equal to . So the probability that the length of AC is less than or equal to must be as well.

Let’s try one more.

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