Question

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

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

Step-by-step solution

- 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.

- 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.

- Compare Quantities A and B

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

Key concepts

Understanding Geometry Problems

Solving geometry problems often starts with understanding the shapes and their properties involved. Knowing how to apply formulas is crucial. These problems can involve various geometric figures like circles, triangles, or polygons. For the current problem, a tangent line and a circle are the focal points.
Your primary approach should involve identifying given information. Here, you know the radius of the circle, one side of the triangle, and the Pythagorean Theorem.

• Identify all parts of the figure
• Look for properties like tangents and radius
• Utilize relevant formulas, like the Pythagorean Theorem, to progress

Remember, visualizing the problem through sketches can help in understanding complex geometry scenarios better.

Right Triangle Essentials

Right triangles are a foundational concept in geometry, characterized by one 90-degree angle. The sides surrounding this angle are called legs, and the side opposite it is the hypotenuse. The Pythagorean Theorem is pivotal in solving right triangles.
In any right triangle, you can express: \[a^2 = b^2 + c^2\]where 'a' is the hypotenuse, and 'b' and 'c' are the other two legs.

• If you know two sides, the third side is easily found using this theorem.
• For special right triangles, like 45-45-90 or 30-60-90, ratios between sides simplify calculations.

The problem given involves a right triangle formed by line AB, line AC, and line BC.

The Role of Tangents and Circles

A tangent is a line that meets a circle at exactly one point. This unique point is called the point of tangency. A vital property about tangents is that they are perpendicular to the radius at the point of tangency.
Understanding this property helps to recognize that whenever a tangent and a radius form an angle, it is a right angle. This fact is the basis for considering a right triangle in this problem.

• Tangents are unique (only one tangent at a point on the circle)
• Tangent-Radius Perpendicularity: key concept for solving related problems

In our situation, line AB is a tangent to the circle at point A, standing perpendicular to radius AC.

Exploring Circle Properties

Circles are symmetrical shapes, characterized by properties like radius, diameter, and circumference. The center of the circle is equidistant from all points on the circle's boundary.
In many geometry problems, especially ones involving tangents, knowing that the radius is perpendicular to tangents can be helpful.

• Radius: constant distance from center to any point on the circle
• Diameter: twice the radius, the longest chord of the circle
• Perpendicular Bisector: the radius to point of tangency

In the given problem, a circle with center C and radius AC serves to establish a right triangle with tangent AB, using the radius and a side of the triangle (BC) effectively.