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Chapter 14: Problem 1

Line \(A B\) is tangent to the circle \(C\) at point \(A\). The radius of the circle with center \(C\) is 5 and \(B C=\) \(\frac{10 \sqrt{3}}{3}\). Quantity A The length of line segment \(A B\) Quantity B The length of line segment \(A C\) a. Quantity A is greater. b. Quantity B is greater. c. The two quantities are equal. d. The relationship cannot be determined from the information given.

Short Answer

Expert verified
After calculation, you will find that Quantity B is greater. Therefore, the answer is \(b. Quantity B is greater\)

Step by step solution

01

- Recall the Pythagorean Theorem

The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This can be written as: \(a^2 = b^2 + c^2\), where a is the hypotenuse and b, c are the lengths of the other two sides.
02

- Apply the Pythagorean Theorem

Here line AB is the hypotenuse, AC is one of the other sides with length 5 (the radius of the circle), and BC is the other side. So we can write the Pythagorean theorem as: \(AB^2 = AC^2 + BC^2 \rightarrow AB^2 = (5)^2 + \left(\frac{10 \sqrt{3}}{3}\right)^2 \). Calculate the value of \(AB^2\) and take the square root to get AB.
03

- Compare Quantities A and B

Finally, compare the length of AB and AC to find which quantity is greater.

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Most popular questions from this chapter

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