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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 \(\mathbf{A}\) 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
b. Quantity B is greater

Step by step solution

01

Understand the given

Let's first understand the problem. A line \(AB\) is tangent to a circle with center \(C\) at point \(A\). The radius \(AC\) of the circle is given as 5. Moreover, the distance \(BC\) from the center of the circle to an external point \(B\) on the line \(AB\) is \(\frac{10\sqrt{3}}{3}\). We need to compare the length of \(AB\) with \(AC\).
02

Employ the Pythagorean Theorem

Since AB is tangent to the circle at point A and AC is the radius at the point of tangency, the triangle ABC is a right triangle with AC as one leg and BC as the hypotenuse. We can therefore apply Pythagoras' theorem which states that in a right-angled triangle, the square of the hypotenuse equals the sum of squares of the other two sides.
03

Calculate Length of AB

Now we can solve for the length of \(AB\). Let's denote the length of AB as \(AB = c\). With \(AC = 5 = a\) and \(BC = \frac{10\sqrt{3}}{3}= b\), using the Pythagorean theorem, we have \(c^2 = b^2 - a^2\). Substituting the given values, we get \(c = \sqrt{(\frac{10\sqrt{3}}{3})^2 - 5^2} = \sqrt{\frac{300}{9} - 25} = \sqrt{\frac{100}{9}} = \frac{10}{3}\).
04

Compare the lengths

We then compare \(AB\) with \(AC\). Our calculated length for AB is \(\frac{10}{3}\) and the length of AC is \(5 = \frac{15}{3}\). Therefore, \(AC>AB\) which means Quantity B is greater than Quantity A.

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Most popular questions from this chapter

\(k>0\) \(l>1\) Quantity \(\mathbf{A}\) \(\frac{1}{\frac{1}{k}+\frac{1}{l}}\) Quantity \(\underline{B}\) \(\frac{k l}{\frac{1}{k}+\frac{1}{l}}\) 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.

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If 16 is the average (arithmetic mean) of \(p, 24,\) and \(q,\) what is \(16(p+q) ?\) a. 180 b. 192 c. 384 d. 524 e. 768
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