Point P is 20 unites from the centre of a circle of radius 29. The number of different chords of the circle through P with inteher lengths is
a] 16
b] 17
c] 32
d] 33
e] 34
help and explaination pls?
I think the answer is (c). First draw a picture of a circle and mark the points P and O(centre). Now convince yourself that the shortest length chord passing through P is the one thats perpendicular to OP.Now compute the length of the chord through Pythogorus theorem. The chord length will be 42. SInce the largest length chord is the diameter, length of the largest length chord through P is 58. Thus the chords range from 42 to 58. Now if you carefully count you will see that the number of chords is two times the number of chords on one side and the answer is 32
Suppose that the circle is centered at the origin (0,0), and that the point P is located at (20,0).
The first chord passing through the point P will be from (29,0) to (-29,0) with a length of 58.
Another chord will be from (-29,0) to (29,0) or in the opposite direction, but will still have the same length of 58.
I suppose that only makes a chord count (with integer length ) of 1 since they both are equal to 58.
That is the longest possible chord.
The shortest possible chord will occur when a point on the circle occurs at (20,y) or (20,-y).
[That creates a right triangle]
The radius (or hypotenuse) is 29,
the base is 20,
the height is y
$\displaystyle y = \sqrt{29^2 - 20^2} = 21$
Thus the chord will be from (20,21) to (20,-21) for an integer length of 42.
That is the shortest chord possible.
The chord length will vary (continously) from 42 to 58. You will have a chords of various lengths between 42 to 58 inclusive.
Simply count the number of integers from 42 to 58 (including 42 and 58).
That should be your answer
Spoiler:
Here's a picture to illustrate the answer. The longest chord (length 58) is the diameter AB. The shortest is the perpendicular chord CD (length 42). These only occur once. Every intermediate length occurs twice, once as X moves along the arc from B to C, and once as X moves along the arc from C to A. The total number is that given by Isomorphism.