A circle with center O has a radius of 17. Chord AB in the same circle has length 30. How far the chord from O?

one of the "standard" right triangles is 8-15-17.

To find the distance between the chord AB and the center O of the circle, we can use the property that the perpendicular bisector of a chord passes through the center of the circle.

Here's how you can find the distance:

1. Draw a line from the center O of the circle to the midpoint M of chord AB. Let's say this line intersects AB at point N.

2. Since the perpendicular bisector of a chord passes through the center of the circle, line MN is perpendicular to chord AB.

3. The length of chord AB is given as 30. Since MN is perpendicular to AB and passes through the midpoint of AB (M), MN is also the height of an isosceles triangle formed by chord AB.

4. In an isosceles triangle, the height (MN) splits the base (AB) into two equal segments. So, AN = NB = 30/2 = 15.

5. Now, we have triangle AON, where AN = 15 and AO = radius of the circle = 17.

6. We need to find the height of triangle AON, which is the distance between the chord AB and the center O.

7. To find the height, we can use the Pythagorean theorem. The theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

8. In triangle AON, AO is the hypotenuse, and AN is one of the perpendicular sides. Using the Pythagorean theorem, we can find the height of the triangle:

AO² = AN² + ON²
17² = 15² + ON²
289 = 225 + ON²
ON² = 289 - 225
ON² = 64

9. Taking the square root of both sides, we get ON = √64 = 8.

Therefore, the distance between chord AB and the center O is 8 units.