Digital SATMath: Geometry and TrigonometryMedium
A city planner is designing a new park. A circular fountain will be placed in the center, and a straight path will run tangent to the fountain. If the fountain has a radius of 5 meters and the point of tangency on the path is 12 meters from the park entrance along the path, what is the straight-line distance from the center of the fountain to the park entrance?
- A7 meters
- B17 meters
- C13 meters
- D169 meters
Show answer & explanationAnswer & explanation
Correct answer: C. 13 meters
A line tangent to a circle is always perpendicular to the radius at the point of tangency. This creates a right-angled triangle. The radius (5m) is one leg, the distance along the path from the tangent point to the entrance (12m) is the other leg, and the distance from the fountain's center to the entrance is the hypotenuse. Using Pythagorean theorem: 5² + 12² = c² -> 25 + 144 = c² -> 169 = c² -> c = √169 = 13 meters.
Why the other options are wrong
- A. This is 12-5, which incorrectly subtracts the radius.
- B. This is 12+5, which incorrectly adds the radius.
- D. This is 13 squared, not the length itself.
Tangent to a Circle
A straight line that touches a circle at exactly one point.
- The tangent line is always perpendicular to the radius at the point of tangency.
- Forms a right angle, useful for Pythagorean theorem and trigonometry.
- Important concept in geometry and calculus.
Memory trick: Tangent means 'touching just once,' and it's always 'right' with the radius.