Use the Pythagorean theorem in relationships between circles and lines, such as chords and tangents.
Chord and Distance from the Center
A perpendicular from the center of a circle to a chord bisects the chord.
For the distance d from the center to chord AB, radius r, and half-chord length l:r² = d² + l²
📘 Example ①
In a circle with radius 10cm, the distance from the center to chord AB is 6cm. Find the length of chord AB.
Answer:l² = 10² - 6² = 100 - 36 = 64. l = 8cm. AB = 2l = 16cm
In a circle with radius 10cm, the distance from the center to chord AB is 6cm. Find the length of chord AB.
Answer:l² = 10² - 6² = 100 - 36 = 64. l = 8cm. AB = 2l = 16cm
💡 Key Point
- Perpendicular from the center to a chord → the midpoint of the chord coincides with the foot of the perpendicular
- Use r² = d² + (AB/2)²
Practice Problem
- In a circle with radius 13cm, the chord length is 24cm. Find the distance from the center to the chord.
Answer and Explanation
- Answer:5cm. d² = 13² - 12² = 169 - 144 = 25. d = 5.